WORKS  OF  PROF.  J.  B.  JOHNSON 

PUBLISHED  BY 

JOHN  WILEY  &  SONS. 


Theory  and  Practice  in  the  Designing  of  Modern 
Framed  Structures. 

4to,  cloth,  $10.00. 
The  Theory  and  Practice  of  Surveying. 

Designed  for  the  use  of  Surveyors  and  Engineers 
generally,  but  especially  for  the  use  of  Students 
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Stadia  and  Earth-work  Tables. 

Including  Four-place  Logarithms,  Logarithmic 
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tions, etc.    8vo,  cloth,  $1.25. 

The  Materials  of  Construction. 

Large  8vo,  800  pages,  650  illustrations,  11  plates, 
complete  index,  $6.00. 

Engineering  Contracts  and  Specifications. 

Including  a  Brief  Synopsis  of  the  Law  of  Contracts 
and  Illustrative  Examples  of  the  General  and  Tech- 
nical Clauses  of  Various  Kinds  of  Engineering 
Specifications.  Designed  for  the  use  of  Students, 
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Publishing  Co.,  Tribune  Building,  New  York. 
Price,  $4.00,  postpaid. 


THE  THEORY  AND  PRACTICE 

OF 

MODERN  FRAMED  STRUCTURES 


DESIGNED  FOR  THE  USE  OF  SCHOOLS, 

AND  FOR 

ENGINEERS  IN  PROFESSIONAL  PRACTICE. 


BY 

J.  B.  JOHNSON,  C.E., 

Professor  of  Civil  Engineering  in  Washington  Universily,  St.  Louis,  Mo.;  Member  of  the  Institution  of  Civil  Engineers; 
Member  of  the  American  Society  of  Civil  Engineers  ;  A/ember  of  the  American  Society  of  Mechanical  Engineers  ; 

etc.,  etc. 

C.  W.  BRYAN,  C.E, 

Engineer  of  the  Edge  Moor  Bridge  Works,  Wilmington,  Del.; 
AND 

F.  E.  TURNEAURE,  C.E., 

Professor  of  Bridge  and  Hydraulic  Engineering,  University  of  Wisconsin,  Madison. 


SIXTH  EDITION,   THOROUGHLY  REVISED. 
FIRST  THOUSAND. 


NEW  YORK: 

JOHN  WILEY  &  SONS. 

London  :  CHAPMAN  &  HALL,  Limited 
1897. 


Copyright,  1893, 

BV 

J.  B.  JOHNSON, 

F.  E.  TURNEAURE, 

C.  W.  BRVAN. 


ROBERT  DROMMOND,  BLHCTROTVPER  AND  PRINTER,  NEW  VORK. 


PREFACE  TO  THE  SIXTH  EDITION. 


While  many  minor  changes  and  corrections  have  been  made  in  each  new  edition  of 
this  work,  these  have  not  been  sufficient  to  warrant  calling  special  attention  to  them.  The 
following  important  additions  have  now  been  made  in  the  sixth  edition : 

(1)  Moment  Tables  for  Cooper's  conventional  method  of  treating  wheel  loads  (Art.  ii2i>). 

(2)  Chapter  IX,  on  Column  Formulae,  has  been  supplemented  by  new  working  formulae 
which  give  correct  working  loads  for  all  lengths  of  column,  and  at  the  same  time  give  greater 
factors  of  safety  on  short  than  on  long  columns,  and  the  reasons  therefor.  This  is  an 
innovation  in  column  formulae,  but  the  authors  feel  that  the  arguments  fully  justify  the 
change.    Diagrams  are  also  given  for  these  new  formulae  which  enable  the  designer  to  take 

/ 

off  his  working  stress  as  soon  as  his  ratio  -  is  approximately  known.    These  are  drawn  for 

all  grades  of  material  from  wrought  iron  to  hard  steel,  or  for  all  "  apparent  elastic  limits  " 
from  30,000  to  50,000  lbs.  per  square  inch. 

(3)  A  new  discussion  of  swing-bridges  (Arts.  175  and  178^),  which  proves  that  the 
ordinary  formulae  are  practically  correct,  since  the  neglecting  of  the  web  system  usually 
compensates  the  errors  made  in  assuming  the  moment  of  inertia  of  the  chord  sections  con- 
stant. The  method  given  in  the  last  edition  of  this  work,  therefore,  and  since  adopted  in 
other  recent  publications  on  drawbridges,  of  considering  the  moment  of  inertia  of  the  chords 
as  variable  and  of  neglecting  the  deflections  due  to  the  web  system,  is  here  shown  to  give 
very  erroneous  results.  These  methods,  therefore,  are  not  only  very  tedious  in  application, 
but  quite  misleading  in  practical  designing. 

July,  1897. 

ill 


PREFACE  TO  THE  FIRST  EDITION. 


It  is  now  less  than  fifty  years  since  the  first  successful  attempt  was  made  to  correctly 
analyze  the  stresses  in  a  framed  structure  and  to  proportion  the  members  to  resist  the  given 
external  forces.*  In  this  comparatively  short  period  the  rational  designing  of  framed 
structures  has  ripened  into  practical  perfection,  and  the  best  current  practice  leaves  little  to 
be  desired  in  the  way  of  further  development.  The  only  material  uncertainties  remaining  are 
the  dynamic  effects  of  moving  loads,  and  these  will  probably  never  submit  themselves  to  any 
very  accurate  determination  or  prediction.  This  would  seem  to  be  a  fitting  time,  therefore, 
for  the  presentation  to  the  engineering  profession  of  a  general  treatise  on  Moder?t  Framed 
Structures. 

The  evolution  of  methods  of  analysis  and  of  construction  has  been  so  rapid  during  this 
generating  period  that  no  sooner  has  a  work  on  structures  appeared  than  it  has  been  found  to 
be  behind  the  current  practice  and  no  longer  representative.  It  is  believed  that  this  rapid 
evolution  of  new  methods  has  about  run  its  course,  and  that  we  have  now  settled  upon  a  line 
of  practice,  both  in  analysis  and  in  construction,  which  will  be  reasonably  fixed  so  long  as  the 
materials  employed  remain  as  they  are  to-day.  It  was  this  conviction  that  led  the  authors  of 
this  work  to  undertake  the  task  of  presenting  the  subject  in  as  concise  and  inclusive  a  form  as 
possible,  to  serve  at  once  the  needs  of  the  student  and  of  the  practitioner. 

This  work  is  something  of  a  compendium,  a  text-book,  and  a  designer's  hand-book,  all  in 
one.  As  a  compendium  it  is  intended  to  cover  a  great  deal  of  ground  without  going  too  much 
into  details  which  are  found  in  standard  works  on  mathematics  and  mechanics.  As  a  text- 
book it  is  intended  to  serve  as  the  student's  manual  in  framed  structures,  after  he  has  had 
a  course  in  mathematics  and  mechanics.  As  a  designer's  hand-book  it  is  intended  to  contain 
such  ready  information  as  any  competent  designer  must  constantly  use,  but  which  he  does  not 
care  to  burden  his  mind  with. 

As  a  text-book  the  discreet  teacher  will  not  undertake  to  go  over  it  all  with  equal  care. 
According  to  the  amount  of  time  he  can  spare  to  this  subject  he  will  use  more  or  less  of  it. 
Chapters  I,  II,  III,  IV,  V,  VIII,  and  IX  are  essential  as  a  ground-work  for  any  intelligent 
designing.  After  these  are  mastered  any  portion  of  the  remainder  may  be  taken  at  pleas- 
ure. It  would  hardly  be  wise,  in  any  case,  to  teach  all  of  Part  I  before  taking  something 
in  Part  II.    After  studying  a  portion  or  all  of  the  seven  chapters  named  above,  it  might  be 


*  By  Mr.  Squire  Whipple,  of  Albany,  N.  Y.    See  foot-note,  p.  8. 


iv 


PREFACE. 


V 


well  to  assign  to  each  student  some  simple  design,  as  of  a  roof  truss  (each  one  taking  a 
different  style  of  truss,  but  all  of  the  same  span,  loads,  spacing,  etc.),  the  teacher  leading  the 
class  in  the  problem,  and  assigning  such  parts  only  of  the  various  chapters  in  Part  II  as  bear 
on  the  several  elements  of  the  design  as  they  arise  for  solution.  This  would  indicate  at  once 
how  Part  II  is  to  be  used  in  actual  designing,  and  it  would  maintain  the  student's  interest  in 
the  theoretical  portion  of  Part  I  by  the  practical  application  of  it. 

If  the  course  is  a  fairly  thorough  one  nearly  all  of  Part  I  should  be  studied  sooner  or  later, 
and  as  much  of  Part  II  as  there  is  time  for.  Probably  in  no  case  would  it  all  be  taught,  but 
the  student,  in  the  various  problems  in  designing  which  are  assigned  to  him,  should  have 
occasion  to  consult  nearly  all  parts  of  the  book. 

The  work  may  be  criticised  on  the  one  hand  for  being  too  concise,  and  on  the  other  for 
being  too  inclusive.  The  authors  have  tried  to  avoid  all  unnecessary  verbiage  and  such 
mathematical  developments  as  are  given  in  works  necessarily  preparatory  to  this,  to  keep  the 
book  from  becoming  too  bulky  ;  and  they  have  intended  to  fairly  cover  the  field  of  structural 
designing  in  which  the  engineer  of  to-day  is  called  upon  to  practise.* 

This  work  has  been  written  by  so  many  persons  that  it  is  only  in  a  limited  sense  that 
those  whose  names  appear  on  the  title-page  may  be  considered  its  authors.  These  latter, 
however,  have  had  the  direction  of  the  work  and  have  written  much  the  larger  portion  of  it. 
It  has  been  their  controlling  motive  to  have  the  book  represent  correctly  the  latest  and  best 
practice,  and  in  many  ways  to  even  point  out  some  improvements  in  both  the  analysis  and  the 
designing  of  structures. 

In  order  that  the  reader  may  always  know  the  particular  author  he  may  be  reading,  the 
following  scheme  is  given  as  a  key  to  such  information.  While  Prof.  Johnson  has  had  general 
charge  of  the  entire  work,  in  an  editorial  capacity,  and  has  written  portions  of  various  chapters 
not  ascribed  to  him,  the  work  has  been  divided  as  follows  : 

Prof.  J.  B.  Johnson,  Chapters  I,  VI,  VIII,  IX,  X,  XI,  XV,  XXIII,  XXV  (Parts  I  and 
III),  and  XXVII. 

Prof.  F.  E.  Turneaure,  Chapters  II,  III,  IV,  V,  VII,  XII  (in  part),  XIII,  and  XIV. 
Mr.  C.  W.  Bryan,  C.E.,  Chapters  XVI,  XVII,  XVIII,  XIX,  XX,  XXI,  XXII,  and  XXV 
(Part  II). 

Mr.  J.  W.  Schaub,  M.  Am.  Soc.  C.  E.,  Chapters  XII  (in  part)  and  XXIV. 

Mr.  David  A.  Molitor,  C.E.,  Chapter  XXVI. 

Mr.  C.  T.  Purdy,  C.E.,  Chapter  XXVIII. 

Mr.  Geo.  H.  Hutchinson,  C.E.,  Chapter  XXIX. 

Mr.  F.  H.  Lewis,  M.  Am.  Soc.  C.  E.,  Appendix  A. 

Mr.  A.  L.  Johnson,  C.E.,  Appendix  B. 

Mr.  Frank  W.  Skinner,  M.  Am.  Soc.  C.  E.,  Appendix  C. 

It  is  only  due  to  Washington  University  to  say  that  at  the  time  he  did  the  v/ork  Prof. 
Turneaure  was  Instructor  in  Civil  Engineering  in  that  institution  (C.E.  Cornell  University), 
while  Messrs.  Schaub^  Bryan,  Molitor,  and  A.  L.  Johnson  are  graduates  from  its  civil  engi- 
neering course. 


*  The  chapter  on  Lock  Gates  which  was  in  the  original  scheme  was  made  unnecessary  by  the  excellent  monograph 
on  this  subject  by  Lieut.  Hodges,  published  in  1892  by  the  Corps  of  Engineers,  U.S.A.,  as  Professional  Papers,  No.  26. 


vi 


PREFACE. 


Mr.  Schaub  has  been  Chief  Engineer  of  two  of  the  largest  bridge  works  of  America, 
namely,  the  Dominion  Bridge  Company  of  Montreal  and  the  Detroit  Bridge  Company.  He 
was  for  many  years  an  assistant  to  Mr.  C.  Shaler  Smith,  one  of  the  great  bridge  engineers  this 
country  has  produced.  He  is  now  General  Manager  of  the  Pottsville  Bridge  Works,  Potts- 
ville,  Pa.    His  chapters  on  draw  bridges  can  therefore  be  regarded  as  authoritative. 

Mr.  Bryan  speaks  also  with  authority,  as  he  has  for  many  years  been  the  Designing 
Engineer  of  the  Edge  Moor  Bridge  Works,  which  are  the  largest  structural  works  in  the 
world. 

Mr.  Purdy  has  designed  many  of  the  tall  steel-skeleton  buildings  of  Chicago,  and  Messrs. 
Hutchinson,  Lewis,  A.  L.  Johnson  and  Skinner  are  also  fully  qualified,  both  theoretically  and 
practically,  to  speak  on  the  subjects  treated  by  them. 

Mr.  Molitor  has  written  a  chapter  in  an  entirely  new  field,  so  far  as  the  English  literature 
is  concerned.  He  has  spent  a  number  of  years  in  Europe  in  engineering  practice,  and  has 
had  an  opportunity  to  cultivate  a  naturally  strong  aesthetic  sense.  His  private  library  and  his 
collections  of  photographs  are  the  main  sources  from  which  he  obtained  his  material.  It  is 
to  be  hoped  that  this  chapter  may  give  an  impetus  to  the  growing  sense  of  dislike  for  the 
innate  ugliness  which  now  characterizes  many  of  the  largest  bridges  of  this  country. 

The  authors  wish  also  to  acknowledge  their  indebtedness  to  Prof.  Green  of  the  University 
of  Michigan,  to  Prof.  Crandall  of  Cornell  University,  to  Prof.  Swain  of  the  Massachusetts 
Institute  of  Technology,  to  Dr.  Eddy,  President  of  Rose  Polytechnic  Institute,  for  many  ideas 
and  methods  which  have  been  incorporated  in  the  body  of  the  work,  and  to  Mr.  Wolcott  C. 
Foster,  for  the  use  of  some  plates  from  his  "  Wooden  Trestle  Bridges."  Other  acknowl- 
edgments will  be  found  in  foot-notes  scattered  through  the  book. 

The  authors  have  spared  no  expense  in  the  matter  of  cuts  and  plates,  nearly  all  of  which 
have  been  specially  drawn  for  this  work,  and  engraved  by  the  American  Bank  Note  Company 
of  New  York.  Only  a  few  of  the  plates  have  been  reproduced  by  photographic  processes. 
The  publisher  of  Hutton's  monograph  of  the  Washington  Bridge  has  kindly  granted  the  use 
of  several  plates  from  that  excellent  work. 

That  this  work  should  fairly  and  adequately  exemplify  the  principles  and  practice  of 
structural  designing  in  America,  and  meet  with  the  approval  of  their  fellow  teachers  and 
practitioners,  has  been  the  constant  hope  and  aim  of 

The  Authors. 

July,  1893. 


TABLE  OF  CONTENTS. 


PART  I. 

THEORY  OF  FRAMED  STRUCTURES. 


CHAPTER  I. 

INTRODUCTORY.     (J.  B.  J.)* 

Fundamental  Definitions— The  Truss  and  its  Elements— Historical  Development  of  the  Truss  Idea, 

CHAPTER  n. 

LAWS  OF  EQUILIBRIUM.     (F.  E.  T.) 

Systems  of  Forces— Resultant  and  Equilibrium  of  Concurrent  Forces— Resultant  and  Equilibrium  of 
Non-concurrent  Forces— Analytical  Applications  of  the  Equations  of  Equilibrium — The  Equilibrium 
Polygon— Laws  of  Equilibrium  applied  to  the  Beam— Applied  to  the  Structure  as  a  Whole— Applied 
to  Single  Joints— Applied  to  Sections— Moment  Diagrams  for  Uniform  Loads   i 

CHAPTER  HL 

ROOF-TRUSSES.     (F.  E.  T.) 

Loads  and  Reactions — Forms  of  Trusses — Analysis  of  a  French  Truss — The  Quadrangular  Truss — 
Analysis  of  a  Crescent  Truss — The  Arch  Truss — Double  Systems  of  Web  Members   3 

CHAPTER  IV. 

BRIDGE-TRUSSES  WITH  UNIFORM  LOADS.     (F.  E.  T.) 

Formulae  and  Diagrams  for  Dead  Load — The  Live  Loads — Shear  and  Bending  Moment  in  a  Beam — 
Chord  Stresses — Web  Stresses  with  Parallel  Chords — The  Warren  Girder  Analyzed — The  Howe 
Truss  Analyzed — The  Pratt  Truss — The  Whipple  Truss  Analyzed — The  Triple  Intersection  Truss 
— The  Post  Truss — The  Baltimore  Truss  Analyzed — Bridge-trusses  with  Inclined  Chords — The 
Parabolic  Bowstring  Truss — The  Double  Bow  or  Lenticular  Truss — The  Pegram  Truss  Analyzed 
— Skew  Bridges. ...   4 

CHAPTER  V. 

BRIDGE-TRUSSES  WITH  WHEEL-LOADS.     (F.  E.  T.) 

Influence  Lines  and  their  Use— Position  of  Moving  Load  for  Maximum  Bending  Moment— Point  of 
Maximum  Moment  in  Plate  Girder— Position  of  Load  for  Maximum  Floor-beam  Concentration— 


*  For  key  to  authors  of  chapters  see  Preface. 


vii 


viii 


CONTENTS. 


Position  of  Load  for  Maximum  Shear — Moment  Tables  for  Wheel-loads — Application  to  a  Pratt 
Truss — Moment  Diagrams — Position  of  Load  for  Maximum  Moment  by  Diagram — Maximum  Shears 
by  Diagram — Computation  of  Panel  Concentrations — Position  of  Loads  for  Maximum  Moments 
and  Shears  on  Trusses  with  Inclined  Chords — Complete  Analysis  of  Truss  with  Inclined  Chords — 
Trusses  with  Double  Systems  of  Web  Members  

CHAPTER  VI. 

CONVENTIONAL  METHODS  OF  ANALYSIS,     (j.  B.  J.) 

Various  Conventional  Methods  of  Treating  Train  Loads — Method  of  Uniform  Train  Load  with  One 
Moving  Concentrated  Load — Method  of  Equivalent  Uniform  Loads — Accuracy  of  Conventional 
Methods — Table  of  Relative  Results  

CHAPTER  VII. 

LATERAL  TRUSS  SYSTEMS.     (F.  E.  T.) 

Stresses  from  Wind  Pressure — Portal  Bracing — Sway  Bracing — Centrifugal  Force  

CHAPTER  VIH. 

BEAMS  AND  CONTINUOUS  GIRDERS.     (J.  B.  J.) 

Historical  Sketch  of  the  Development  of  the  True  Theory  of  Beams — Elementary  Principles — Table  of 
Moments  of  Inertia  and  Moments  of  Resistance — Stress-strain  Diagrams — Distribution  of  Stress 
after  the  Elastic  Limit  is  Passed — Rational  Equations  for  the  Moment  of  Resistance  at  Rupture — 
To  Find  the  Moment  of  Inertia  of  a  Section  composed  of  Rectangles — To  Find  the  Centre  of 
Gravity  and  Moment  of  Inertia  of  any  Section— General  Relation  between  Shear  and  Bending 
Moment  in  Beams — Deflection  of  Beams — Distribution  of  Shearing  Stress  in  Beams — Continuous 
Girders — Bending  Moments  at  Supports— Shears  at  Supports  

CHAPTER  IX. 

COLUMN   FORMUL.C.     (j.  B.  J.) 

Crushing  Strength — Three  Methods  of  Column  Failure — Effect  of  End  Conditions — A  New  Formula — 
Experimental  Tests  on  Full-sized  Columns — Johnson's  Straight-line  Formula— Working  Formulae 
for  Different  Materials  

CHAPTER  X. 

COMBINED  DIRECT  AND  BENDING  STRESSES. — SECONDARY  STRESSES,     (j.  B.  J.) 

Action  of  Direct  and  Bending  Stresses — General  Formula  for  Maximum  Fibre  Stress — Examples  Solved 
— Bending  Action  on  Columns  Fixed  at  One  End — The  Trussed  Beam — Gravity  Lines  not  Meeting 
at  a  Point — Members  not  Loaded  on  Centre  of  Gravity  Lines  

CHAPTER  XI. 

SUSPENSION- BRIDGES.     (j.  B.  J.) 

Introduction — Theory  of  Suspension-bridges — Stiffening  Truss  for  Partial  Loads — Maximum  Shear — 
Maximum  Bending  Moment — Action  of  Stays — Distribution  of  the  Loads  over  the  Different 
Systems    


CONTENTS. 


ix 


CHAPTER  XII. 

.    SWING  BRIDGES.    (F.  E.  T.  AND  J.  W.  S.) 

PACB 

General  Formulae — Constants  for  Reactions — Centre-bearing  Pivot  with  Three  Supports — Numerical 
Example — Rim-bearing  Turn-table  with  Four  Supports — Equal  Moments  at  Middle  Supports — 
Rim  bearing  with  Three  Supports— Lift  Swing  Bridges — Wind  Stresses — Moment  of  Inertia  Variable.  179 

CHAPTER  Xin. 

CANTILEVER  BRIDGES.     (F.  E.  T.) 

Varieties  of  Cantilevers — Analysis  of  Stresses — Indiana  and  Kentucky  Bridge     197 

CHAPTER  XIV. 

ARCH  BRIDGES.     (F.  E.  T.) 

General  Principles— Application  of  Equilibrium  Polygon — Positions  of  Loads  for  Maximum  Stresses — 
Deflection  of  Curved  Beams — Fundamental  Equations — Parabolic  Arch  of  Two  Hinges  and  Variable 
Moment  of  Inertia — Computation  of  Stresses — Temperature  Stresses — Parabolic  Arch  with  Fixed 
Ends — Computation  of  Stresses — Temperature  Stresses   203 

CHAPTER  XV. 

DEFLECTION  OF  FRAMED  STRUCTURES  AND  DISTRIBUTION  OF  LOADS  OVER  REDUNDANT  MEMBERS. 

(J.  B.  J.) 

General  Formula  for  Deflection  of  a  Framed  Structure — Deflection  Formulae  for  a  Pratt  Truss — Numeri- 
cal Example— Effect  of  Height  on  Deflection — Inelastic  Deflection  -Camber — Errors  caused  by 
Neglecting  Deflection  due  to  Web  Distortions — Numerical  Computation  of  Deflection — Determina- 
tion of  Stresses  in  Redundant  Members— Direct  Measurement  of  Bridge  Strains   219 


}  PART  II. 

/  STRUCTURAL  DESIGNING. 

CHAPTER  XVI. 

STYLES  OF  STRUCTURES  AND  DETERMINING  CONDITIONS.     (C.  W.  B.) 

General  Consideration— Proper  Structure  to  Use  at  a  Given  Crossing— The  Current  Styles— Bridge 
Floors — Riveted  Truss  or  Lattice  Bridges — Pin-connected  Truss  Bridges   233 

CHAPTER  XVII. 

DESIGN  OF  INDIVIDUAL  TRUSS  MEMBERS.     (C.  W.  B.) 

The  Fatigue  of  Metals  and  the  New  Method  of  Dimensioning — The  Usual  Method — Tension  Members 
— Compression  Members    242 


CHAPTER  XVIII. 

DETAILS  OF  CONSTRUCTION.     (C.  W.  B.) 

Riveting — Sizes,  Spacing-Strength  of  Riveted  Joints— Watertown  Arsenal  Tests — Designation  and 
l,ocation  of   Rivets— Anchor  Bolts— Pins— Bending  Moments  and  Stresses  on  Pins  -Bearing 


X 


CONTENTS. 


Strength  of  Rollers — Experimental  Researches — Provisions  for  Expansion — Sub-panelling — 
StifTened  Tension  Members — Floor  Systems — Lateral  Systems— Portal  and  Sway  Bracing— Chord 
Joints — Camber — Lattice  Bars    257 

CHAPTER  XIX. 

THE  PLATE  GIRDER.     (C.  W.  B.) 

The  Moments  and  Shears — The  Web  and  its  Splicing — Web  Stiflfeners — Flange  Areas — Distribution  of 
Rivets  in  Flanges — The  Detail  Design  of  a  Deck  Plate  Girder — The  Economic  Depth — Flange  Areas 
and  Length  of  Plates — Thickness  of  Web  and  Size  of  Flange  Angles — Transfer  of  Loads  and  Stresses 
— Stresses  on  Rivets — Other  Details  of  the  Design — Design  of  Through  Plate  Girder  Spans — Details 


of  the  Design  ; .  292 

CHAPTER  XX. 

ROOF  TRUSSES.     (C.  W.  B.) 

Styles  of  Trusses — Coverings — Loads — Allowed  Stresses — Economic  Spacing  of  Trusses — The  Purlins — 
Detail  Design  of  a  Sixty-foot  Span — The  Bracing  of  Roofs   313 

CHAPTER  XXL 

THE  COMPLETE  DESIGN  OF  A  SINGLE  TRACK  RAILWAY  BRIDGE.     (C.  W.  B.) 

Data — Allowed  Stresses — Tabulation  of  Stresses  and  Sectional  Areas — Detail  Design  of  Each  Member — 
Design  of  the  Iron  Floor  System— The  Lateral  Bracing — The  Portal  Strut — Joint  Details — The 
Pins — The  Pin  Plates — The  Top  Chord  Joints — The  End  Shoes — Other  Details   321 

CHAPTER  XXn. 

HIGHWAY    BRIDGES.     (C.  W.  B.) 

Tne  Loads — Styles  of  Floors — The  Panel  Length — Design  of  the  Floor — Design  of  the  Trusses   343 

CHAPTER  XXHL 

THE  DETAIL  DESIGN  OF  A  HOWE  TRUSS  BRIDGE,     (j.  B.  J.) 

The  Howe  Truss — Chord  Splices — Angle  Blocks  and  Bearings — Various  Details — Working  Stresses — 

Weights  and  Quantities — Long  Span  Howe  Truss  Bridges   349 

CHAPTER  XXIV. 

THE  DESIGN  OF  SWING  BRIDGES.     (j.  W.  S.) 

Plate  Gi<der  Swing  Bridges — Riveted  Pony  Truss  Swing  Bridges — Truss  Swing  Bridges — Standard 
Forms— Supports  at  Centre — End  Lifting  Arrangements — Tlie  Ram  with  Toggle  Joints — Machinery 
for  Operating — Resistances  Evaluated — Summary  of  Work  Done — Detail  Design  of  a  2i6-foot  Span,  357 


CHAPTER  XXV. 

TIMBER  AND  IRON  TRESTLES  AND  ELEVATED  RAILROADS,     (j.  B.  J.  AND  C.  W.  B.) 

Timber  Trestles — The  Framed  Bent — Bearing  Joints — Floor  Systems — Sway  Bracing — Longitudinal 
Bracing — Bent  and  Post  Spiices — Working  Stresses — Iron  Trestles — General  Design — Lateral 
Stability  and  Batter  of  cJoiumns — Length  of  Tower  Span  and  Longitudinal  Stability — Economic 
Length  of  Intermediate  oi  Variable  Span — Design  of  the  Columns — Lateral  and  Longitudinal 
Bracing — Stresses  in  the  'l"owers — Details  of  the  Towers — Elevated  Railroads — Characteristic 
Features — The  Live  Loads — Lateral  and  Longitudinal  Stability — Solution  of  the  General  Case  of 


CONTENTS. 


xi 


Horizontal  Forces  acting  on  Columns  Fixed  at  the  Ground — Selected  Examples — Economic  Span 
Lengths  and  Approximate  Cost   382 

CHAPTER  XXVI. 

THE  ESTHETIC   DESIGN  OF  BRIDGES.     (D.  A.  M.) 

Hindrances  to  Esthetic  Design— Fundamental  Principles — Influence  of  Building  Material,  Color,  and 
Shades  and  Shadows  on  the  Appearance  of  a  Bridge — Ornamentation — Esthetic  Design — Sub- 
structure— Superstructure — Roadway — Railings — Discussion  of  the  Plates   411 

CHAPTER  XXVIl. 

STAND-PIPES  AND  ELEVATED  TANKS.  (J.B.J.) 

Use  of  Water  Towers — Stand-pipes — Dimensions  and  Thickness  of  Metal — Wind  Moment  and  Anchor- 
age— The  Anchorage  Brackets — Details  of  Construction — Ornamentation — Elevated  Tanks — Design 
of  Tank — The  Trestle  Tower — Specification  for  Plate  Material   427 

CHAPTER  XXVIII. 

IRON  AND  STEEL  TALL  BUILDING  CONSTRUCTION.     (C.  T.  P.) 

Modern  Building  Construction — Arrangement  of  Columns  and  Beams — Economy  of  Wide  Spacing 
of  Floor-beams — Calculation  of  Column  Loads — Design  of  Foundations — Design  of  Spandrel 
Sections — Dimensioning  of  Beams  and  Columns — Wind  Bracing   439 

CHAPTER  XXIX. 

IRON  AND  STEEL  MILL-BUILDING  CONSTRUCTION.     (G.  H.  H.) 

General  Types  of  Mill-buildings — The  External  Forces — Methods  of  Analysis — Action  of  the  Horizontal 
Forces  on  the  Columns — Systems  of  Bracing — Types  of  Roof  Trusses — Columns  and  Girders  for 
Travelling-cranes— Details  of  Construction — Analysis  for  Wind  Forces— Analytical  and  Graphical 


Processes   460 

APPENDICES. 

Appendix  A. — Structural  Steel  and  General  Specifications  (F.  H.  L.)  475 

Appendix  B. — Processes  in  the  Manufacture  and  in  the  Inspection  of  Iron  and  Steel 

Structures  (A.  L.  J.)   501 

Appendix  C— American  Methods  of  Erection  of  Bridges  and  Structures  (F.  W.  S.)   508 


THEORY  AND  PRACTICE 


IN  THE  DESIGNING  OF 

MODERN  FRAMED  STRUCTURES. 


Part  I. 

THEORY  OF  FRAMED  STRUCTURES. 

CHAPTER  I. 
DEFINITIONS  AND  HISTORICAL  DEVELOPMENT. 

1.  A  Simple  Framed  or  Articulated  Structure  is  one  composed  of  straight  members 
so  attached  at  their  extremities  as  to  cause  the  structure  to  act  as  one  rigid  body.  It  may  be 
contrasted  with  masonry  structures  on  the  one  hand  and  with  soHd  beams,  plate  girders, 
arches,  wire  suspension-bridges,  and  the  like  on  the  other. 

2.  The  External  Forces  include  all  the  loads  and  foundation  reactions,  including  the 
weight  of  the  structure  itself,  which  act  upon  and  which  tend  to  distort  it.  These  forces  are 
always  replaced  by  their  equivalent  forces  applied  at  the  joints  before  the  direct  stresses  in 
the  members  can  be  computed. 

3.  Strain  is  the  distortion  of  a  body  caused  by  the  application  of  one  or  more  external 
forces.*  It  is  measured  in  units  of  length,  as  inches,  and  not  in  pounds.  The  proportional, 
or  relative,  strain  is  usually  meant,  this  being  the  distortion  per  unit  of  original  length,  or  in 
other  words,  the  actual  distortion  divided  by  the  original  length  of  the  member. 

4.  Stress  is  the  resistance  of  a  body  to  distortion,  and  can  only  exist  in  unconfined  bodies 
when  these  are  solid  or  plastic.  It  is  measured  in  pounds  or  tons  the  same  as  the  external 
forces.  The  stresses  resist,  or  hold  in  equilibrium,  the  external  forces,  but  the  immediate 
cause  of  the  stress  is  the  distortion  of  the  body.f  The  external  forces  upon  a  framed  struc- 
ture distort  the  members  until  the  resisting  stresses  developed  in  them  are  sufficient  to  hold 
in  equilibrium  these  external  or  distorting  forces.  For  bodies  in  equilibrium  the  external 
forces  and  the  internal  stresses  stand  in  the  relation  to  each  other  of  action  and  reaction  in 
mechanics.  Furthermore,  when  the  stress  in  one  member  is  resisted  by  or  transmitted  to 
another  member  or  part  of  a  structure  it  acts  upon  the  latter  as  an  external  force.  Thus  the 
reaction  of  the  foundation  is  a  stress  in  the  masonry  support,  but  is  to  be  treated  as  an 
external  force  acting  upon  the  superposed  structure. 

5.  Relation  between  Stress  and  Strain, — In  all  solid  bodies  there  is  a  definite  relation 
between  the  intensity  of  the  stress  and  the  amount  of  the  accompanying  strain.  No  body 
is  so  rigid  as  to  remain  unstrained,  or  undistorted,  under  the  application  of  any  finite  external 

*  In  popular  language  "strain"  and  "stress"  are  often  confused  and  used  indiscriminately.  Some  authors  of 
repute  have  also  followed  the  popular  usage,  but  the  definitions  here  given  conform  to  the  practice  of  the  leading 
authorities. 

I  It  is  common  to  say  the  distortion  is  caused  by  the  stress.  But  a  resistance  cannot  be  the  cause  of  the  thin^ 
resisted.    Though  coincident  in  time  and  place,  the  distortion  is  really  the  cause  of  the  stress. 


2 


MODERN  FRAMED  STRUCTURES. 


force,  however  small.*  Within  a  certain  limit  for  any  particular  material  a  given  increase  in 
the  external  force  is  always  accompanied  by  a  proportionate  increase  in  the  strain,  or  distor- 
tion, and  this  develops  a  like  increase  in  the  stress,  or  resistance.  Thus  if  any  bar  of  rolled 
iron  or  steel  be  distorted  by  an  external  force  (pull  or  thrust)  of  28  lbs.  per  square  inch,  it 
will  stretch  or  shorten,  as  the  case  may  be,  an  amount  equal  to  one  one-millionth  part  of  its 
length,  the  internal  stress,  or  resistance  to  distortion,  then  coming  to  be  just  equal  to  the 
external  force  of  28  lbs.  per  square  inch.  An  external  force  of  28,000  lbs.  per  square  inch 
distorts  or  strains  the  bar  one  one-thousandth  part  of  its  length  and  then  develops  in  the  bar 
a  resistance  or  stress  of  28,000  lbs.  per  square  inch.  It  is  evident,  however,  that  this  resist- 
ance cannot  continue  to  increase  indefinitely  in  proportion  to  the  distortion.  There  always 
comes  a  time,  if  the  external  force  continues  to  increase,  when  a  greater  increment  of  distor- 
tion is  requisite  to  develop  a  given  increment  of  resistance.    This  point  is  called 

6.  The  Elastic  Limit. — Below  this  limit  the  stress  and  the  strain  are  proportional,  equal 
increments  of  one  always  producing  equal  increments  of  the  other.f  Also,  below  this  limit, 
when  the  distorting  force  ceases  to  act  the  body  returns  to  its  original  shape  and  dimensions 
and  the  stress  is  relieved.  If  the  body  be  distorted  beyond  the  elastic  limit,  the  strain 
increases  more  rapidly  than  the  stress,  or  than  the  external  force,  these  two  always  of  necessity 
being  equal  to  each  other,  and  some  of  the  distortion  becomes  permanent.  That  is,  when 
the  external  force  is  removed  the  body  does  not  fully  return  to  its  original  dimensions,  but 
remains  permanently  distorted  somewhat,  or  it  is  said  to  have  "  taken  a  set." 

7.  The  Modulus  of  Elasticity  is  the  ratio  of  the  stress  per  unit  of  area  to  the  relative 
strain,  or  distortion,  which  accompanies  it.  In  other  words,  it  is  unit  stress  divided  by  unit 
strain,  or 

unit  stress  _f  _fl, 
~  unit  strain  ~'  oi~  a' 
/ 


where  a  =  distortion  or  strain  (either  elongation  or  compression); 

/  =  original  length  of  part  under  stress  ; 

f  =  stress  per  unit  area  (pounds  per  square  inch  in  English  units). 

Since  pounds  and  inches  are  the  standards  used  in  English,  the  modulus  of  elasticity  as 
given  and  used  in  all  English  works  must  be  understood  to  represent  pounds  per  square  inch, 
the  denominator  of  our  fraction  in  eq.  i  being  an  abstract  number.:}: 

This  modulus  or  ratio  is  constant  within  the  elastic  limit.  Beyond  that  it  steadily 
decreases  until  it  reduces  to  zero  in  the  case  of  solid  metals  where  the  material  becomes 
plastic  and  draws  out  or  compresses  under  a  constant  load.  All  working  stresses  are,  or 
should  be,  well  within  the  elastic  limit,  and  hence  for  all  such  stresses  this  ratio  is  constant 
for  any  given  material.  When  it  is  known  the  resisting  stresses  can  be  found  for  a  known 
distortion,  or  the  distortion  may  be  computed  for  a  known  stress.  The  determination  of  this 
ratio  requires  very  delicate  measuring  apparatus  with  the  most  careful  and  expert  handling. 
Tabular  values  given  for  these  moduli  for  different  materials  in  standard  works  are  not  very 
reliable.  Thus,  for  all  the  rolled  irons  and  steels  this  modulus  is  remarkably  constant,  being 
perhaps  always  between  26,000,000  and  31,000,000  lbs.  per  square  inch  for  the  ordinary 


*  It  may  be  found  helpful  to  think  of  all  engineering  materials  as  composed  of  india-rubber  in  order  to  free  our 
minds  from  the  notions  of  absolute  rigidity  which  are  apt  t«  be  associated  with  the  harder  kinds  of  structural  materials. 

f  This  is  known  as  Hooke's  Law  and  was  originally  expressed  by  the  Latin  phrase  "  [/I  iensio  sic  vis." 

X  If  this  denominator  could  become  unity,  which  it  never  can  in  solids,  then  the  fraction,  or  E,  would  represent  the 
number  of  pounds  per  square  inch  required  to  stretch  a  body  to  twice  its  original  length,  and  the  modulus  of  elasticity 
is  sometimes  so  defined. 


DEFINITIONS  AND  HISTORICAL  DEVELOPMENT. 


3 


temperatures,  while  the  tensile  strength  of  these  metals  will  vary  from  45,000  lbs.  per  square 
inch  for  wrought-iron  and  soft  steel  to  over  200,000  lbs.  per  square  inch  for  hard-drawn  steel 
wire.  It  is  an  extremely  valuable  property  of  engineering  materials,  and  is  used  to  great 
advantage  by  the  scientific  designer. 

8.  Examples. — The  following  examples  are  given  to  illustrate  some  of  the  uses  to  be  made  of  the 
modulus  of  elasticity.  In  solving  these  problems,  take  the  modulus  of  wrought-iron  as  27,000,000,  and  of 
steel  as  28,500,000;  of  cast-iron  as  12,000,000;  and  of  timber  as  1,500,000  lbs.  per  square  inch. 

1.  A  steel-wire  cable  5  miles  lon<;  and  one  square  inch  in  solid  section  is  pulled  with  an  average  force 
of  15,000  lbs.    What  is  the  strain,  or  stretch  ? 

2.  The  rim  of  a  cast-iron  fly  wheel  10  feet  in  diameter  is  subjected  to  a  tensile  stress  of  5000  lbs.  per 
square  inch  from  the  centrifugal  force.    How  much  is  its  diameter  increased? 

3.  If  an  iron  or  steel  rail  30  feet  long  is  prevented  from  expanding,  what  will  be  the  stress  in  it  per 
square  inch  resulting  from  a  rise  of  temperature  of  80°  F.,  the  coefficient  of  expansion  being  taken  at 
0,0000065  ?    t4JM.        ,£5-^  a^fg^fe^  ^^^•^I'l^ftr 

4.  A  series  of  wooden  posts  superposed  upon  each  other  in  a  building  to  a  total  height  of  60  feet  are 
subjected  to  an  average  compressive  stress  of  1000  lbs.  per  square  inch.  How  much  will  be  the  settlement 
at  the  top  from  this  cause  ? 

THE  TRUSS  AND  ITS  ELEMENTS. 

9.  A  Truss  is  a  framed  or  jointed  structure  designed  to  act  as  a  beam  while  each 
iflcmber  is  usually  subjected  to  longitudinal  stress  only,  either  tension  or  compression. 

10.  The  Struts  are  those  members  which  are  compressed  endwise,  and  which  therefore 
have  developed  in  them  compressive  resistances  or  stresses.  Struts  are  sometimes  called 
Po.sts,  or  Columns. 

11.  The  Ties  are  those  members  which  are  extended,  and  which  thus  have  developed  in 
them  tensile  resistances  or  stresses. 

12.  The  Upper  and  Lower  Chords  are  composed  of  the  upper  and  lower  longitudinal 
members  respectively.  When  the  loads  are  downward  and  the  truss  is  supported  at  its  ends, 
the  upper  chord  is  always  in  compression  and  the  lower  chord  always  in  tension.  The  spaces 
between  the  chord  joints  are  called  panels. 

13.  The  Web  Members  are  those  which  join  the  two  chords.  They  are  alternately  in 
tension  and  compression,  or  the  struts  and  ties  alternate  in  the  web  systein. 

14.  A  Counterbrace  is  a  member  which  is  designed  to  resist  both  tensile  and  compressive 
strains.  That  is,  for  one  position  of  the  load  the  member  may  be  elongated,  while  for 
another  it  may  be  compressed,  and  hence  at  different  times  it  must  resist  both  extension  and 
compression.  When  two  or  more  external  forces  act  upon  it,  some  of  which  tend  to  compress 
the  member  and  others  to  extend  it,  it  is  evident  that  only  the  algebraic  sum  of  these  forces 
really  acts  upon  the  member.  It  is  subjected  to  a  stress,  therefore,  equal  to  and  of  the  same 
kind  as  the  algebraic  sum  of  all  the  external  forces  acting  upon  it. 

Hence,  also,  a  tension  member  or  tie  may  resist  a  compressive  external  force  without 
becoming  a  counterbrace,  so  long  as  this  compressive  force  is  smaller  than  another  extending 
force  which  also  acts.  Similarly  with  .struts,  they  can  resist  'ensile  external  forces  without 
becoming  ties,  so  long  as  these  are  less  than  other  compressive  forces  which  continue  active. 
The  residual  stress  is  tension  in  the  one  case  and  compression  in  the  other,  for  which  only  the 
member  is  designed,  and  therefore  we  may  say  that  both  struts  and  ties  may  resist  the  contrary 
external  forces  without  becoming  counterbraces,  the  stresses  in  the  member  always  being 
of  one  sign  *  or  kind. 

*  If  this  book  tension  is  called  minus  and  compression  plus.  There  is  no  objection  to  following  the  contrary  rule. 
It  is  only  important  that  both  stresses  and  forces  of  opposite  kinds  should  enter  with  opposite  signs. 


4 


MODERN  FRAMED  STRUCTURES. 


15.  Mains  and  Counters. — A  main  member,  whether  strut  or  tie,  is  one  which  acts 
when  the  entire  structure  is  loaded.  A  counter  is  one  which  acts  only  for  particular  partial 
loads. 

2   4  6  8  10  12  14  ^6  W  » 


Fig.  I. 

16.  Illustration. — In  Fig.  i,  which  represents  a  Pratt  Truss,  the  compression  members, 
or  struts,  are  shown  by  double  lines,  and  the  tension  members,  or  ties,  by  single  lines.  When 
the  end  posts  are  inclined  as  in  the  figure,  they  would  seem  to  belong  about  as  much  to  the 
upper  chord  as  to  the  web  system.  They  are  usually  spoken  of  separately  as  the  "  inclined 
end-posts  "  or  the  "  batter-braces." 

The  members  7-10,  9-12,  lo-ii,  and  12-13  ^•'^  counters,  or  counter-ties.  There  are  no 
counter-braces  in  this  truss ;  that  is,  no  members  have  to  resist  both  tension  and  compression. 

The  tension  members  2-3,4-5,6-7,  8-9,  11-14,  13-16,  15-18,  and  17-20  are  main  tie- 
rods.  The  members  1-2  and  19-20  are  not  elements  of  the  truss  proper,  since  they  serve 
only  to  carry  the  loads  at  I  and  at  19  to  the  hip-joints. 

17.  The  Action  of  a  Truss. — Since  a  truss  is  a  jointed  structure  composed  of  rigid  but 
elastic  members  so  arranged  as  to  form  an  unyielding  combination,  it  must  be  composed  of 
an  assemblage  of  rigid  polygons.  But  the  only  rigid  polygonal  figure  is  a  triangle.  A  truss 
must  therefore  be  composed  of  an  assemblage  of  triangles.  Any  assemblage  of  triangles 
fastened  together  at  their  apices,  consecutive  figures  having  sides  in  common,  is  a  truss,  and 
will  act  as  a  beam.  In  Fig.  l  the  triangles  are  all  right-angled.  A  load  placed  at  joint  7,  for 
instance,  is  carried  by  the  truss  as  a  beam  to  the  abutments  at  O  and  21.  The  part  of  this 
load  which  goes  to  the  left  abutment  may  be  conceived  as  being  carried  up  to  6,  down  to  5, 
up  to  4,  down  to  3,  up  to  2,  and  then  down  to  o,  where  it  passes  to  the  ground.  The  part 
which  goes  to  the  right  passes  over  the  path  7,  10,  9,  12,  1 1,  14,  13,  16,  15,  18,  17,  20,  and  21. 
Thus  for  such  a  load  the  ties  7-10  and  9-12  are  put  under  stress,  while  8-9,  lo-ii,  and  12-13 
are  idle,  as  well  as  the  post  7-8  and  the  hangers  1-2  and  19-20.  If  all  these  idle  members 
were  removed,  the  truss  would  stand  under  this  particular  loading,  since  it  would  still  remain 
an  assemblage  of  triangles,  properly  joined.  If  the  load  were  placed  at  9,  9-8  and  9-12  would 
be  under  stress,  while  7-10  and  lo-il  would  be  idle.  If  the  two  middle  joints  9  and  11  were 
loaded  equally,  the  part  of  the  load  at  9  going  to  the  right  is  just  balanced  by  the  part  of  the 
load  at  1 1  going  to  the  left,  and  hence  there  is  no  stress  in  the  intermediate  web  members 
9-12,  lo-ii,  9-10,  and  11-12.    The  counter-ties  7-10  and  12-13  are  also  idle. 


Fig.  2. 


In  Fig.  2  we  have  generalized  conceptions  of  a  truss.  They  are  assemblages  of  triangles, 
adjacent  figures  having  a  common  side,  and  exemplify  the  generic  idea  of  a  truss. 


DEFINITIONS  AND  HISTORICAL  DEVELOPMENT 


5 


A  truss  is  not  weakened  from  its  want  of  symmetry  or  from  its  sagging  in  the  middle, 
provided  all  the  members  are  properly  proportioned  to  carry  their  loads. 

A  Through-bridge  is  one  in  which  the  roadway  is  carried  directly  at  the  bottom-chord 
joints,  with  lateral  bracing  overhead  between  the  top-chord  joints,  thus  enclosing  a  space 
through  which  the  load  passes. 

A  Deck-bridge  is  one  in  which  the  roadway  is  carried  directly  at  the  top-chord  joints,  or 
on  the  upper  chords  themselves.  The  trusses  are  usually  placed  closer  together  than  on 
through-bridges,  the  roadway  extending  over  them. 

A  Pony  Truss  is  a  low  truss  of  short  span,  with  the  roadway  carried  at  the  bottom  joints, 
but  not  of  sufificient  height  to  allow  of  the  upper  lateral  bracing.  The  trusses  are  stayed,  or 
held  to  place,  by  bracing,  connected  with  the  floor  system. 

HISTORICAL  DEVELOPMENT  OF  THE  TRUSS  IDEA. 

l8.  Primitive  Systems. — The  earliest  forms  of  truss  were  built  of  timber,  the  progres- 
sive development  of  the  forms  used  for  bridge  purposes  being  shown  in  the  following  figures. 


Figs,  "^a,  3^,  y  are  three  forms  of  truss  construction  employed  by  Palladio,  a  famous 
Italian  architect,  1560-80.  These  trusses  were  built  entirely  of  timber,  and  are  believed  to 
be  the  earliest  examples  of  a  scientific  use  of  the  truss  element,  the  rigid  triangle.  Palladio 
wrote  an  elaborate  illustrated  treatise  on  architecture  in  which  these  and  other  forms  of 
truss  have  been  preserved. 

The  mastery  of  the  principles  of  truss  construction  did  not  follow  the  practice  of  Palladio, 
and  so  fine  an  example  of  the  use  of  the  truss  element  is  not  found  again  for  nearly  three 
hundred  years. 

Fig.  4  represents  one  span,  170  feet  long,  of  a  bridge  over  the  Rhine  at  Schaffhausen, 
built  in  1758  by  Ulric  Grubenmann,  a  carpenter  by  trade  but  really  a  great  engineer.  He 

/ 


6 


MODERN  FRAMED  STRUCTURES. 


afterwards  built  a  wooden  bridge  of  similar  design,  366  feet  long,  near  Baden.  Both  these 
bridges  served  their  purpose  till  destroyed  by  Napoleon  in  1799. 


Fig.  4. 


Fig.  5  represents  the  "Permanent  Bridge"  over  the  Schuylkill  River  at  Philadelphia, 
built  by  Mr.  Timothy  Palmer  of  Newburyport,  Mass.,  in  1804.*  The  middle  span  was  195  feet 
and  the  side  spans  were  150  feet  each.  It  was  covered  in  and  continued  in  service  till  1850, 
when  it  was  replaced  by  a  bridge  for  railroad  purposes. 


FiQ.  5. 

In  1804  Mr.  Theodore  Burr  built  a  bridge  over  the  Hudson  River  at  Waterford,  in  four 
spans  of  154,  161,  176,  and  180  feet  clear  span,  respectively,  after  the  pattern  shown  in  Fig.  6. 


Ail  members  were  of  timber,  and  counter-struts  were  inserted  the  entire  length,  thus  giving 
great  rigidity.  This  is  probably  the  most  scientific  design  for  an  all-wooden  bridge  ever 
invented,  and  for  a  half-century  it  stood  unrivalled  for  cheapness  and  efificiency  for  highway 
purposes  in  this  country.  These  bridges  were  always  covered  in,  the  covering  extending 
many  feet  beyond  the  end  of  the  truss  proper. 

In  Fig.  7  is  shown  a  view  of  the  Colossus  Bridge  over  the  Schuylkill  River  at  Philadel- 
phia, built  in  1812  by  Mr.  Lewis  Wernway.  It  was  340  feet  clear  span,  and  marked  a  great 
advance  on  previous  practice  in  America  in  the  length  of  span.  It  was  destroyed  by  fire  in 
1838.  These  three  gentlemen,  Palmer,  Burr,  and  Wernway,  were,  up  to  1840,  the  leading 
bridge  engineers  of  America. 

All  the  above  types  of  bridges.  Figs.  4-7,  are  composite  forms  and  not  simple  trusses. 


*  See  Paper  on  American  R.  R.  Bridges,  by  Theodore  Cooper,  Trans.  Am.  Soc.  Civ.  Engrs.  Vol.  XXI,  p.  i. 


DEFINITIONS  AND  HISTORICAL  DEVELOPMENT. 


7 


The  last  three  are  really  trussed  arches,  the  arch  carrying  the  greater  part  of  the  load,  while 
the  truss  prevented  undue  distortions  in  it.  The  Burr  truss  would  have  had  considerable 
strength  alone  if  the  bottom  chord  had  been  more  efificiently  spliced. 


19.  Early  Forms  of  Simple  Trusses. — It  is  believed  that  the  Howe  truss.  Fig.  8,  is 
the  earliest  foi  ni  of  simple  truss  for  long  spans  ever  built  for  bridge  purposes,  except  those  of 


Palladio.*  It  was  patented  in  the  United  States  in  1840  by  William  Howe.  Although  but  a 
modification  of  the  Burr  truss,  it  was  designed  to  carry  the  entire  load  as  a  truss,  without  any 
aid  from  arches  or  from  auxiliary  struts  projecting  out  from  the  abutments.  It  is  constructed 
of  timber  except  the  vertical  tie-rods,  which  are  of  iron.  It  has  been  repeated  thousands  of 
times  in  this  country  on  both  railroads  and  highways,  and  will  long  continue  a  favorite  form 
of  bridge  for  new  lines  of  railroad,  and  for  wagon-roads  in  places  where  timber  is  cheap  and 
the  site  distant  from  a  railway  station.    It  will  be  fully  treated  in  the  second  part  of  this  work. 

The  earliest  type  of  iron  truss  construction  in  the  United  States  was  the  invention  of 
Mr.  Wendell  Bollman,  Fig.  9.     It  is  really  a  trussed  beam,  each  joint-load  being  carried 


directly  to  the  ends  of  the  upper  chord  by  two  inclined  tension  members.  There  is  no  stress 
in  the  lower  chord.  It  is  therefore  a  kind  of  multiple  suspension  system  and  not  properly  a 
truss  at  all.    It  was  largely  used  from  1840  to  1850  on  the  Baltimore  and  Ohio  Railroad. 

An  improvement  on  the  Bollman  truss,  as  shown  in  Figs.  10  and  lo«,  was  made  about  185 1 
by  Mr.  Albert  Fink,  then  assistant  engineer  on  the  B.  &  O.  R.  R.  These  are  called  "  Fink 
trusses,"  the  loads  being  carried  at  the  lower  and  upper  joints  respectively.  This  design  has 
been  more  often  used  for  deck-bridges,  though  many  through-spans  have  been  built  on  this 
plan. 


*  Long's  truss  was  braced  out  from  the  abutments  and  hence  did  not  give  simple  veitical  reactions. 


8 


MODERN  FRAMED  STRUCTURES. 


In  both  the  Bollman  and  the  Fink  trusses  the  compression  members  were  usually  made 
of  cast-iron,  and  in  neither  case  was  there  any  stress  in  the  lower  chord  for  stat  ic  loads. 
Bollman  trusses  were  very  soon  abandoned,  but  Fink  trusses  were  built  for  some  twenty  five 


Fig.  io. 


Fig.  io<7. 


years,  or  down  to  about  1876,  the  compression  members  being  made  finally  of  wrought-iron. 
Pins  and  eye-bars  were  freely  but  not  exclusively  used  for  joints  in  the  Bollman  and  Fink 
trusses. 


f= — \ 

\ 

\ 

r  ^ 

Fig.  II. 


The  first  near  approach  to  the  modern  style  of  iron  truss-bridges  was  made  by  Squire 
Whipple  *  in  1852,  in  a  bridge  of  146  feet  span,  which  he  built  7  miles  north  of  Troy,  N.  Y., 


*  Mr.  Squire  Whipple,  C.E.,  a  philosophical-instrument  maker  of  Utica,  N.  Y.,  seems  to  have  the  distinguished 
honor  of  being  the  first  man  who  ever  correctly  and  adequately  analyzed  the  stresses  in  a  truss,  that  is,  in  a  frame- 
work by  which  loads  are  carried  horizontally  from  joint  to  joint  by  means  of  chord  and  web  systems  and  finally 
delivered  vertically  upon  the  abutments.  This  is  the  true  function  of  a  truss;  and  although  Palladio,  Long,  Howe,  and 
Pratt  had  already  designed  and  built  such  structures,  they  had  never  known  what  stresses  the  members  were  subjected 
to,  and  did  as  all  engineers  and  builders  did  in  those  days— dimensioned  their  parts  in  accordance  with  such  experience 
and  judgment  as  they  could  bring  to  bear  upon  the  problem.    For  instance,  the  vertical  tie-rods  in  the  Howe  truss 


DEFINITIONS  AND  HISTORICAL  DEVELOPMENT. 


9 


on  the  Rensselaer  and  Saratoga  Railway  (Fig.  1 1).  The  ties  in  the  web  system  extend  over  two 
panels,  and  it  is  therefore  called  a  "  double-intersection  "  truss.  The  lower  chord  was  com- 
posed of  wrought-iron  links  passing  over  wrought-iron  trunnions  in  the  bottoms  of  the  posts. 
All  the  compression  members  were  of  cast-iron,  and  it  was  pin-connected  in  both  upper  and 
lower  chords.    This  form  of  arrangement  of  members  is  still  known  as  the  Whipple  truss. 

In  1863  Mr.  John  W.  Murphy  first  used  wrought-iron  for  all  the  compression  members  in 
truss  construction,  but  still  used  cast-iron  in  joint-blocks  and  pedestals.  On  account  of  this 
improvement  by  Mr,  Murphy  in  the  Whipple  truss,  the  modern  wrought-iron  or  steel  double- 
intersection,  horizontal  chord,  truss  is  sometimes  called  the  Murphy-Whipple  truss. 

In  1861  Mr.  J.  H.  Linville  first  used  wide  forged  eye-bars  and  wrought-iron  posts  in  the 
web  system.    He  still  retained  the  cast-iron  upper  chord. 

To  Messrs.  Whipple,  Murphy,  and  Linville,  therefore,  is  largely  due  the  credit  for 
establishing  in  this  country  the  distinctive  practice  of  eye-bar  and  pin  connections  which  are 
still  used  here  on  all  long-span  iron  bridges. 


Fig.  12. 


From  1865  to  1880  a  great  many  railroad  and  highway  bridges  were  built  under  patents 
granted  to  Mr.  S.  S.  Post,  all  being  of  the  style  shown  in  Fig.  12.  This  truss  is  known  as  the 
Post  truss,  its  distinctive  feature  being  that  the  web  struts,  instead  of  standing  vertically,  have 
a  horizontal  run  of  one  half  a  panel  length,  while  the  ties  have  a  horizontal  run  of  one  and 
one-half  panel  lengths.  The  theoretical  economy  from  this  arrangement  is  now  thought  to  be 
offset  by  corresponding  practical  disadvantages  and  the  truss  is  no  longer  built. 

Since  this  historical  account  treats  only  of  truss  forms,  no  mention  is  made  of  the  early 
cast-iron  arch-bridges,  and  of  iron-link  and  wire  suspension-bridges,  many  kinds  of  which,  of 
long  spans,  preceded  the  introduction  of  the  truss  proper.  In  fact,  in  England  and  on  the 
Continent  the  truss  developed  out  of  a  combination  of  the  arch  and  suspension  systems,  cast- 
iron  being  used  in  the  arched  upper  chord,  and  wrought-iron  links  in  the  curved  lower  chord, 
the  two  being  rigidly  held  with  vertical  struts  and  diagonal  tie-rods. 

Since  about  1870  cast-iron  has  been  entirely  abandoned  in  America  in  the  construction  of 
railroad  bridges,  and  since  about  1880  in  highway  bridges  as  well. 

were  at  first  made  of  the  same  size  from  end  to  end.  Mr.  Whipple's  work  is  preserved  in  a  small  book  of  120  pp. 
entitled  "  A  Work  on  Bridge  Building,  consisting  of  Two  Essays,  the  one  elementary  and  general,  the  other  giving 
Original  Plans  and  Practical  Details  for  Iron  and  Wooden  Bridges.    By  S.  Whipple,  C.E.    Utica,  N.  Y.,  1847." 

This  is  a  remarkable  work.  The  author  not  only  has  correctly  analyzed  bridges  for  both  sialic  and  moving  loads, 
correctly  dimensioning  all  members,  including  the  counters,  but  he  computes  the  total  "strain-lengths"  of  various 
styles  of  bridges  and  compares  their  relative  weights  in  this  manner.  He  also  gives  a  very  good  column  formula, 
finds  the  best  ratio  of  panel  length  to  height  of  truss,  and  of  height  of  truss  to  length,  and  compares  the  relative  cost 
of  wood  and  iron  bridges.  There  are  ten  plates  of  details,  including  his  first  designs  for  the  double-intersection  iron 
truss  since  called  by  his  name.  His  methods  of  analysis  were  graphical  but  strictly  correct.  He  was  the  first  to  use 
pin-connections  in  the  bottom  chord,  and  his  designs  for  this  wrought-iron  pin-joint  are  extremely  interesting. 

This  book  had  been  published  three  or  four  years  when  Hermann  Haupt  wrote  his  work  on  bridges.  Apparently, 
Mr.  Haupt  had  never  seen  a  copy  of  it,  since  he  claims  his  work  as  original  also,  and  there  is  no  internal  evidence 
that  he  had  seen  Whipple's  book.    His  methods  of  analysis  are  much  cruder  than  Mr.  Whipple's  and  far  less  complete. 

The  theory  of  the  stone  arch,  and  of  arch  and  suspension  bridges  under  fixed  or  uniform  loads,  was  early 
developed,  but  the  true  theory  of  truss  action  seems  to  have  originated  wiih  Mr.  Whipple.  His  manuscript  was 
written  in  1846.  ^e.^  Development  of  the  Iron  Bridge,  hy  S.  Whipple  in  A'.  R.  Gazette,  Apr.  19,  1889;  and  Discussion  by 
A.  P.  Boiler  in  Transactions  Am.  Soc.  Civ.  Engrs.  Vol.  XXV,  p.  362.  Also  American  R.  R.  Bridges,  by  Theodore 
Cooper,  Trans.  Am.  Soc.  Civ.  Engrs.  Vol.  XXI,  p.  I. 


lO 


MODERN  FRAMED  STRUCTURES. 


The  favorite  style  of  truss  now  for  moderate  spans  for  all  purposes  is  the  Pratt  truss 
(Fig.  i).  It  was  patented  in  1844  by  Thomas  W.  and  Caleb  Pratt  as  a  combination  wood  and 
iron  bridge.  It  was  a  variation  from  the  Howe  truss  in  that  the  diagonals  were  of  iron  and 
used  in  tension,  while  the  verticals  were  struts  and  were  made  of  timber.  It  never  became  a 
popular  style  until  wrought-iron  came  to  be  used  exclusively  in  bridge  construction,  when  it 
was  found  to  have  advantages  over  all  other  forms.  It  will  be  fully  developed  in  the  body  of 
this  work.* 

In  closing  this  short  account  of  the  development  of  the  idea  of  the  simple  truss,  it  should 
be  said  that  only  within  the  last  twenty-five  or  thirty  years  have  the  mathematical  principles 
governing  the  distribution  of  stresses  in  a  truss  been  generally  understood,  and  for  a  still 
shorter  period  has  the  actual  strength  of  full-sized  members  and  joints  been  even  approxi- 
mately known.  All  the  earlier  examples  of  bridge  construction  were  designed  and  executed 
by  carpenters  and  mechanics  wholly  ignorant  either  of  the  values  of  the  stresses  or  of  the 
strength  of  the  parts,  except  as  experience  had  educated  their  judgments  of  what  would 
probably  serve  the  purpose.  They  are  deserving,  therefore,  of  high  honor  for  the  great  works 
they  were  able  to  build  without  any  of  those  scientific  aids  now  offered  to  every  student  in 
our  numerous  engineering  schools. 


*  No  historical  account  of  American  iron  bridges  would  be  complete  without  some  notice  of  the  efforts  of  Thomas 
Paine  to  introduce  long  cast-iron  arch-bridges  of  low  rise.  As  early  as  1786  he  advocated  the  use  of  cast-iron  for  long 
arch-bridges,  with  a  rise  of  about  one  twentieth  of  the  span ;  and  he  had  such  arches  made  and  tested  at  his  own  expense 
90  feet  in  length,  as  models  for  a  400-foot  span  which  he  urged  Congress  to  build  as  an  example  to  educate  the  public. 
The  Academy  of  Sciences  of  Paris  reported  favorably  on  his  design  for  this  length  of  span,  but  his  model  was  sold  for 
debt  and  afterwards  used  in  England.    He  never  took  out  a  patent,  his  object  being  purely  benevolent. 


APPLICATION  OF  THE  LAWS  OF  EQUILIBRIUM. 


II 


CHAPTER  II. 

APPLICATION  OF  THE  LAWS  OF  EQUILIBRIUM  TO  FRAMED  STRUCTURES. 

DEFINITIONS. 

20.  Forces  are  concurrent  when  their  lines  of  action  meet  in  a  point  ;  non-concurrent 
when  their  lines  of  action  do  not  so  meet. 

Forces  may  also  be  coplanar,  that  is,  lying  in  the  same  plane ;  or  non-coplanar,  lying  in 
different  planes.    Coplanar  forces  only  will  be  here  considered. 

A  force  is  fully  defined  when  its  ainonnt,  its  direction,  and  its  position  are  known. 

21.  The  Moment  of  a  Force  about  a  point  is  the  product  of  the  force  into  the  perpen- 
dicular distance  from  the  point  to  the  line  along  which  the  force  acts ;  it  is  a  measure  of  the 
rotative  action  of  the  force  about  the  point. 

22.  A  Couple  is  a  pair  of  equal  and  opposite  forces  having  dif?erent  lines  of  action. 

23.  Equilibrium. — A  system  of  forces  acting  upon  a  body  is  in  equilibrium,  or  balanced, 
when  the  state  of  motion  of  the  body  is  not  thereby  changed  ;  e.g.,  a  body  at  rest  or  moving 
at  a  uniform  velocity  is  being  acted  upon  by  a  balanced  system  of  forces.  The  body  also  is 
said  to  be  in  equilibrium. 

As  we  distinguish  two  kinds  of  motion,  translation  and  rotation,  so  we  may  distinguish 
two  kinds  of  equilibrium,  equilibrium  of  translation  and  equilibrium  of  rotation.  A  body  to 
be  in  complete  equilibrium  must  be  so  in  both  these  senses,  and  one  does  not  imply  the 
other. 

24.  The  Resultant  of  a  system  of  forces  is  a  single  force  which  will  replace  that  system 
as  regards  its  effect  upon  the  state  of  motion  of  the  body  acted  upon.  A  force  equal  and 
opposite  to  the  resultant  will  balance  the  resultant  and  therefore  the  original  system.  In  the 
single  case  where  the  system  reduces  to  a  couple  no  one  force  will  replace  the  system. 

For  equilibrium  of  translation  the  resultant  must  equal  zero.  For  equilibrium  of  rotation 
the  sum  of  the  moments  of  the  forces  about  any  point  must  equal  zero. 

RESULTANT  AND  EQUILIBRIUM  OF  CONCURRENT  FORCES. 

25.  Graphically. — Let  /',.../',,  Fig.  13  («),  be  a  system  of  concurrent  forces  applied  at 
A.    Their  resultant  may  be  found  as  follows  : 

Lay  off  0\,  Fig.  13  (b),  equal  by  scale  to  P,  and  having  the  same  direction,  then  from  I 
lay  off  1-2  equal  and  parallel  to  P^.  By  the  principle  of  the  parallelogram  of  forces,  O2  is  the 
resultant  of  P^  and  in  amount  and  direction.  Similarly,  by  laying  off  2-3,  3-4,  and  4-5 
equal  and  parallel  to  P^,  ,  and  ,  respectively,  we  have  6>3  equal  to  the  resultant  of  , 
,  and  P, ;  C4  the  resultant  of  P, ,  P, ,  P, ,  and  P, ;  and  finally  6*5  equal  to  th«  resultant  of 
our  given  system  in  amount  and  direction.    Its  point  of  application  is  A. 

The  order  in  which  the  forces  are  laid  off  in  ib)  is  immaterial,  as  each  force  will  evidently 
have  the  same  effect  upon  the  final  position  of  a  point  following  around  the  figure  in  whatever 
part  of  the  path  the  force  occurs.  Thus  the  order  P, ,  P, ,  P,,  P,,  and  P,  gives  the  figure 
Oi'2'^'4S.  We  must,  however,  be  careful  to  draw  each  force  in  its  true  direction;  e.g.,  P, 
must  be  drawn  from  2  towards  3,  and  not  from  2  towards  3". 


MODERN  FRAMED  STRUCTURES. 


A  figure  such  as  Fig.  13  (5)  is  called  a  Force  Polygon. 

Since  R  is  the  resultant  of  .  .  .  P^,  if  we  apply  a  force  R'  at  A,  equal  and  opposite  to 
R,  the  forces  R',  P^  .  .  .  P^  will  form  a  balanced  system.    This  is  seen  to  be  true  from  the 


Fig.  13. 


force  polygon  also,  for  since  5(9  is  equal  and  parallel  to  R',  the  resultant  of  the  system 
P^...P^,  R'  is  zero,  and  therefore  we  have  equilibrium  of  translation.  Equilibrium  of 
rotation  is  not  in  question,  the  forces  being  concurrent.  Expressed  graphically,  the  only 
condition  necessary  for  equilibrium  of  a  system  of  concurrent  forces  is  that  their  force  polygoji 
must  close. 

To  sum  up,  we  may  state  that  the  resultant  of  a  system  of  concurrent  forces  is  given  in 
amount  and  direction  by  the  closing  line  of  the  force-polygon,  this  closing  line  being  drawn 
from  the  origin  to  the  end  of  the  last  force  ;  and  the  force  necessary  to  balance  the  given 
system  is  given  by  this  same  closing  line  drawn  in  the  opposite  direction.  The  resultant  is 
simply  the  shortest  way  of  passing  from  the  origin  of  our  roundabout  system  to  its  end.  Any 
system  of  concurrent  forces  with  its  force  polygon  beginning  at  O  and  ending  at  5  would  be 
equivalent  to  the  given  system,  and  if  beginning  at  5  and  ending  at  O  would  balance  the 
system.  Moreover,  each  of  the  forces  in  a  closed  polygon  is  equal  and  opposite  to  the 
resultant  of  all  the  other  forces, 

26.  Algebraically. — Take  the  same  system  of  forces  as  before,  Fig.  14.    Resolve  each 

force  into  horizontal  and  vertical  components  (any  two 
directions  at  right  angles  will  do  as  well),  or  along  the 
axis  of  Xand  the  axis  of  F  respectively.  These  com- 
ponents form  a  system  equivalent  to  the  given  system. 
Let  hor.  comp.  be  the  algebraic  sum  of  the  horizontal 
components,  =  //  in  the  figure,  and  2  vert.  comp.  that 
of  the  vertical  components,  =  Fin  figure.  The  result- 
ant, R,  in  amount  is  evidently  equal  to 


V'(^  hor.  comp.)"  -|-  (2  vert.  comp.)'. 


The  tangent  of  the  angle  between  R  and  the  axis  of  F  is 

2  hor.  comp.        ,    ,       „  .    ,         .     ,  • 

given  by  -^^^  — ,  and  thus  R  is  determmed  in 

■'2:  vert.  comp. 

amount  and  direction.    Its  point  of  application  is  A  as 

before,  whence  it  becomes  fully  known. 


APPLICATION  OF  THE  LAWS  OF  EQUILIBRIUM.  13 

For  equilibrium,  R  must  be  zero,  or  hor.  comp.)'  -f-       vert,  comp.)'  =  o,  which 

requires  that 

'2  hor.  comp.  =0,  (i) 

and 

"2  vert.  comp.  =  0,  (2) 

These  two  equations  express  algebraically  the  condition  we  have  already  expressed  graphically 
by  requiring  that  the  force  polygon  must  close.  Referring  to  Fig.  13  {b\  we  see  that  2  hor. 
comp.  is  the  net  horizontal  displacement  in  passing  from  O  to  5,=  B^,  and  that  2  vert.  comp. 
is  the  net  vertical  displacement,  =  OB  ;  also  that  R  —  s/i^.  hor.  comp.)'  -|-  {2  vert,  comp.)'. 
For  equilibrium,  R  =  O. 

27.  Remarks. — The  foregoing  conditions  of  equilibrium  are  precisely  similar  to  the 
conditions  necessary  to  a  balanced  land-survey.  As  there,  the  plot  must  close,  or  the 
latitudes  and  departures  must  each  sum  up  zero,  so  here,  our  force  polygon  must  close,  or 
2  hor.  comp.  =  o  and  2  vert.  comp.  =  o.  And  further,  as  we  can  supply  two  unknown  quan- 
tities in  the  former  case,  so  here,  having  a  system  in  equilibrium,  we  can,  either  graphically  or 
by  the  use  of  the  above  equations,  compute  two  unknowns.  They  may  be  either  the 
amounts  or  directions  of  two  forces,  the  amount  of  one  and  the  direction  of  another,  or  the 
amount  and  direction  of  one. 


RESULTANT  AND  EQUILIBRIUM  OF  NON-CONCURRENT  FORCES. 


Fig.  15. 


28.  Graphically. — Let  /',.../',,  Fig.  15,  be  a  system  of 
non-concurrent  forces.  Required  their  resultant  and  condi- 
tions of  equilibrium. 

We  can  combine  with  P, ,  getting  their  resultant  R, ,  then 
this  resultant  with  Pg,  getting and  finally with  ,  getting 
R,,  the  resultant  of  the  entire  system  in  amount,  direction,  and 
position.  To  save  drawing  the  separate  parallelograms  we  may 
construct  a  force  polygon  as  in  Fig.  16  (b),  and  draw  the  hues 
O2,  O^,  and  O4,  these  being  the  resultants.  R,  ,  R,,  and  R,,  in 
amount  and  direction.  Then  from^i,  the  intersection  of  P^  and 
P,,  draw  AB  parallel  to  R^,  and  from  the  intersection  of  AB 

with  P,  draw  BC  parallel  to  R,,  and  lastly  CD  parallel 
to  .^3.  The  line  CD  will  be  the  line  of  action  of  R^ 
whose  amount  and  direction  are  given  in  the  force 
polygon. 

I        I  /    V  2p  The  ?\gvne  A  BCD  \?,  caWed  an  equilihriuvi  polygon, 

/Pj  /        *D     B;^  ■      and  AB,  BC,  and  CD  are  its  segments.    The  point  O 

is  called  the  pole,  and  R,,  R,,  and  R^  are  rays,  of  the 
force  polygon  ;  Oi  2  3  4  is  called  the  load  line. 

To  cause  equilibrium  there  must  be  a  force  R^ , 
equal  and  opposite  to  R^  and  applied  in  the  line  CD. 
Graphically  then,  for  equilibrium  of  non-concurrent 
forces,  the  force  polygon  must  close,  giving  equilib- 
rium of  translation,  and  the  last  force  must  coincide  ivith  the  last  segment  of  the  equilibrium 
polygon,  giving  equilibrium  of  rotation.    It  is  important  to  remember  that  each  segment  of  an 
equilibrium  polygon  is  the  line  of  action  of  the  resultant  of  all  the  forces  to  the  left  of  that 


Fig.  16. 


14 


MODERN  FRAMED  S7 RUCTURES. 


segment;  and  if  the  system  is  a  balanced  one,  it  is  as  well  the  line  of  action  of  the  resultant 
of  all  the  forces  upon  eitlicr  side  of  that  segment.  When  the  forces  are  parallel  or  nearly  so, 
a  special  expedient  is  necessary  to  enable  us  to  draw  an  equilibrium  polygon.  This  case  is 
treated  of  in  Arts.  37-44. 

29.  Analytically. — Resolve  the  forces  into  horizontal  and  vertical  components,  as  in  the 

case  of  concurrent  forces.    The  amount  of  the  resultant, 
R,  is  given  by  \^  {'E  hor.  comp.)'  +  (.5'  vert,  comp,)' ;  its 
"2  hor.  comp. 


direction  by 


Its  line  of  action  is  found 


'2  vert.  comp. 

by  putting  its  moment  about  A,  equal  to  the  sum  of 
the  moments  of  the  given  forces,  or  Ra  =  2  mom. 
To  cause  equilibrium  we  must  have  R,  or 


Fig.  17. 

moments,  or  2  mom., 
equilibrium : 


V  {2  hor,  comp.)*  -f-  {2  vert,  comp.)',  =  o, 

giving  equilibrium  of  translation,  and  the  sum  of  the 
O,  giving  equilibrium  of  rotation.    This  gives  us  three  equations  of 


2  hor.  comp.  =  o, 


(3) 


2  vert.  comp.  =  o,  . 
2  mom.         =  o,  . 


 (4) 

 (5) 


which  hold  true  for  any  system  of  non-concurrent  forces  in  equilibrium,  and  hence,  having 
such  a  system,  we  can  in  general  determine  three  unknowns.  As  the  forces  are  as  a  rule 
known  in  position  and  direction  from  other  considerations,  the  unknowns  are  usually  the 
amounts  of  three  forces.  If,  however,  the  three  unknown  forces  themselves  meet  in  a  point, 
they  are  indeterminate  ;  for,  since  the  system  is  by  supposition  in  equilibrium,  the  resultant  of 
the  known  forces  must  pass  through  the  same  point,  and  the  system  is  for  our  purposes  a  con- 
current one,  and  we  have  but  tzi'o  independent  equations.  Instead  of  the  first  two  equations 
above,  it  is  often  more  convenient  to  write  two  other  moment  equations,  taking  a  new  centre 
of  moments  each  time.    Equations  (3)  and  (4)  are  then  not  independent. 


APPLICATION   OF   THE    EQUATIONS   OF  EQUILIBRIUM. 

30.  Methods. —  In  determining  the  stresses  in  framed  structures  there  are  three  general 
methods  of  applying  the  equations  of  equilibrium. 

1st.    To  the  structure  as  a  whole. 
2d.  To  any  single  joint. 

3d.  By  passing  a  section  through  the  structure,  removing  one  portion  and  applying  the 
equations  of  equilibrium  to  the  remaining  portion,  the  stresses  in  the  members  cut  being 
replaced  by  equal  external  forces.    This  is  known  as  the  method  of  sections. 

I.  Analytical  Application. 

31.  First.  To  the  Structure  as  a  Whole. — The  external  forces  acting  upon  a  structure 
in  equilibrium  form  a  balanced  system,  to  which  may  be  applied  the  three  equations  of 
equilibrium  of  Art.  29.  We  can  therefore  in  general  fully  determine  these  external  forces, 
provided  there  are  not  more  than  three  unknown. 


APPLICATION  OF  THE  LAWS  OF  EQUILIBRIUM. 


Example  i.  Suppose  the  roof-truss  in  Fig.  i8  to  be  acted  upon  by  the  wind-pressure, 
W,  acting  normally  to  the  roof;  the  weight,  G,  of  the  roof  and  truss  applied  at  their  centre 
of  gravity  and  acting  downwards  ;  and 
the  abutment  reactions  as  yet  un- 
known in  direction  or  amount.  These 
comprise  all  the  external  forces. 

Fig.  19  shows  the  truss  free  from 
the  abutments,  with  the  abutment 
reactions,  R'  and  R" ,  put  in.  The 
left  end  of  the  truss  is  supposed  to  rest  upon  rollers,  the  abutment  at  B  taking  all  the 
horizontal  thrust  due  to  wind.  This  being  the  case,  R'  will  be  vertical  and  R"  inclined  at 
some  angle  Q  with  the  vertical.  The  unknowns  are  R' ,  R",  and  0.  Applying  our  three  equa- 
tions of  equilibrium  :  2  hor.  comp.  =  o  gives 

Wsma-  R"  s\nB  =  o\  {a) 


2  vert.  comp.  =  O  gives 

R  +  R"  cos  d  -  Wcos  a  -  G  =  0  \ 


2  mom.  about  B  =  o  gives 


From  {c) 

From  (a) 

From  {6)  and  {d) 


R'  = 


Wa  +  ^Gl 


{d) 


R"  s\n  e  =  W  sin  a  .   ,    .  {e) 

Wa  +  ^Gl 


R"  cose=  lVcosa-\-G 


R" 


sin 


cos 


therefore  R',  R",  and  6* 


and  since  R"  =  VR'"  sin'  6  +  R'"'  cos'  e,  and  tan  6*  r= 
are  readily  found. 

In  the  above  example  we  have  for  convenience  called  forces  to  the  right,  and  upwards, 
and  moments  tending  to  produce  rotation  with  the  hands  of  a  watch,  />osiih'e ;  and  those  in 
the  opposite  directions,  negative.  The  opposite  convention  would  do  as  well,  the  only  thing 
necessary  being  to  introduce  opposite  forces  and  opposite  moments  with  unlike  signs. 

Example  2.  Bridge-truss  (Fig.  20)  with  three  loads,  each  =  10,000  lbs.  Required  the 
abutment  reactions.    Rollers  at  A.    Weight  of  truss,  G,  =  30,000  lbs. 


 90  

Fig.  21. 


B 


Fig.  21  shows  the  truss  free  with  all  external  forces  put  in.  There  being  rollers  at  A, 
R'  will  be  vertical ;  R"  will  make  some  angle  0  with  the  vertical.    It  will  usually  be  more 


i6 


MODERN  FRAMED  STRUCTURES. 


convenient  to  deal  with  the  hor.  and  vert,  components  of  forces  whose  directions  are 
unknown,  the  unknowns  then  being  these  components  in  amount.  Our  unknowns  are  thus 
R ,  Rh' t  and  Ry'.    Applying  now  our  equations  of  equilibrium  :  2  hor.  comp.  =  o  gives 


This  result  might  readily  have  been  foreseen  by  inspection,  there  being  no  other  horizontal 
force.  In  arriving  at  conclusions  by  inspection  we  must,  however,  be  very  careful  to  see  that 
tiiey  are  based  upon  some  one  of  the  three  conditions  of  equilibrium,  and  where  the  result 
is  doubtful  we  should  always  return  to  the  rigid  method,— consider  the  structure  by  itself,  put 
in  all  forces,  and  write  out  in  detail  the  equations  of  equilibrium. 

Returning  to  the  example  : — An  equation  of  moments  about  5  as  a  centre  will  give  R' 
directly,  after  which  Ry'  can  be  found  by  a  second  moment  equation  with  A  as  centre,  or  by 
'2  vert.  comp.  =  o. 

Example  3  (Fig.  22).  Hinges  at  A,  B,  and  C.  Loads  P,  and  P^;  weight  of  structure 
neglected.    What  are  the  reactions  at  A  and  B,  also  the  hinge  reaction  at  6"  ?    In  neglecting 


the  weight  of  the  structure  the  problem  is  to  find  that  portion  of  the  reactions  due  to  the 
loads  /*,  and  P^.  The  portion  due  to  the  weight  may  be  found  separately  and  combined  with 
the  other,  giving  the  total  reactions. 

There  being  a  hinge  at  C,  allowing  one  part  of  the  structure  to  turn  freely  upon  the 
other,  we  have  two  separate  framed  structures,  AC  and  CB,  to  which  may  be  applied  the 
conditions  of  equilibrium.    Fig.  23  shows  each  structure  separated  and  the  external  forces 


—  Rh"  =  O,    or    Rh"  =  O. 


c 


Fig.  22. 


Fig.  23. 


put  in,  the  vert,  and  hor.  components  of  the  unknown  forces  being  indicated.  At  C  we 
have  a  simple  case  of  action  and  reaction, — the  forces  acting  upon  the  two  portions  are  equal 


APPLICATION  OF  THE  LAWS  OF  EQUILIBRIUM. 


17 


and  opposite.  We  have  now  six  unknown  quantities  and  can  write  three  equations  for  each 
structure.    For  the  left-hand  structure,      vert.  comp.  =  o  gives  us 

Ry'  -P,-Ry"'  =  0;  {a) 

2  hor.  comp.  =  o  gives 

R^'-R„"'  =  o;  {l>) 

2  mom.  about  C  =  o  gives 

R/  Xy-P,Xc,-  R,:  Xh  =  o.  ic) 

For  the  right-hand  structure,  2  vert.  comp.  =  o  gives 

Ry"  -^Ry"  -  P,  =  o;    ........  (^) 

2  hor.  comp,  =  o  gives 

R„"'-R,r  =  0'.  {e) 

2  mom.  about  C  =  o  gives 

P,Xc,  +  R„"xk-R^^"xil  =  o  (/) 

These  six  equations  are  readily  solved  for  the  six  unknowns.  From  (7;)  and  {e)  we  have  at 
once,  Rf/  =  Rh'"  =  Rh">  a  result  evident  from  inspection. 

In  this  example  let  P,  =  lOOO  lbs.,  P^  =  2000  lbs.,  /  =  lOO  ft.,  i\  =  10  ft.,  c,  =  15  ft.,  and 
k  =  40  ft.    What  are  the  numerical  values  of  the  reactions? 

Note. — The  value  of  Ry"  should  come  out  =  —  300  lbs.  The  minus  sign  merely  indi- 
cates that  the  direction  of  Ry"  has  been  wrongly  assumed  ;  it  should  act  upivards  on  the 
portion  AC. 

The  above  problem  may  be  solved  more  directly  as  follows:  Since  the  entire  structure, 
ACB,  is  in  equilibrium,  our  equations  will  apply  as  well  to  it.  The  external  forces  are  R^^, 
RAP„P,,Ri/',^x^dRy".    2  mom.  about  A  —  o  gives 

P.  X  (i/  -  O  +  P,X  (*/  +  c,)  -  Ry"  X  /  =  O, 
from  which  Ry'  is  found.    2  vert.  comp.  =  o  gives 

Ry'  +  Ry"  -P^-P^=0, 

from  which  we  get  Ry  at  once.  To  get  Rh,  Rr"  ,  Ry",  and  Ru",  we  must  pass  to  the  single 
structure^  AC,  or  CB,  whence  these  unknowns  are  readily  found. 


Fig.  24. 

Example  4.  Cantilever  bridge.  Joints  at  ^4,  /?,  C  and  Z).  Loads  as  shown  ;  weight  of  bridge  neglected. 
P  =  10,000  lbs.    Find  reactions  at  C,  D,  E,  and  F.    Notice  that  here  we  have  three  independent  structures. 


i8 


MODERN  FRAMED  STRUCTURES. 


32.  Second.    To  Single  Joints  to  Find  Stresses. — Since  all  parts  of  a  structure  at 

rest  are  in  equilibrium,  we  may  evidently  apply  the  laws  of  equilibrium  to  the  forces  acting 
upon  any  portion  of  that  structure.  That  portion  may  be  a  single  joint,  a  single  member  or 
part  of  a  member,  or  it  may  include  several  joints  and  members.  The  forces  acting  upon  the 
portion  may  be  part  external  forces  and  part  internal  forces  or  stresses,  or  they  may  be  wholly 
stresses.    Fig.  25  illustrates  five  different  portions  of  the  roof-truss,  to  which  may  be  applied 

the  equations  of  equilibrium.  The  letters  ,  5',,  etc.,  denote 
forces  due  to  stresses  in  the  members  cut ;  the  direction  in 
which  they  act  along  the  members,  whether  towards  or  away 
from  the  portion  considered,  is  known  only  after  solution. 

Example  i.  Let  it  be  required  to  find  the  stresses  in  all 
the  members  of  the  above  truss ;  loads  as  shown. 

We  must  first  find  the  abutment  reaction  at  A  or  B,  treat- 
ing the  structure  as  a  whole.  ^  mom.  about  B  =  o  gives 
R'l-  Wl-  Wx¥=  o,  from  which  R'  =  ^W. 

In  treating  now  of  single  joints  we  are  dealing  with  con- 
current forces,  hence  we  have  but  two  independent  equations 
at  our  disposal  and  can  therefore  find  but  two  unknown  forces. 
After  finding  R'  there  are  but  two  unknown  forces  acting  at 
A,  viz.,  the  stresses  in  ^dTand  AD.  Separating  this  joint.  Fig. 
25  {b),  and  replacing  these  stresses  by  the  forces  5,  and  5,.  as- 
suming them  to  act  as  shown,  we  are  ready  to  apply  our  equa- 
tions of  condition.    '2  vert.  comp.  =  o  gives 

R'  -  W  -  S,cose-\-  S,  cos  a  =  o. 

2  hor.  comp.  =  o  gives 

S,  sin  a  —  S,  sin  0  =  o. 

From  these  two  equations  S,  and  5,  are  easily  found.  If  the  result  in  either  case  is  nega- 
tive, then  we  have  assumed  our  force  in  the  wrong  direction.  Now  S^  being  the  force  exerted 
upon  the  lower  end  ol  AC  by  the  upper  end,  the  direction  of  the  arrow  indicates  compression  in 
AC.   Similarly,  AD  is  in  tension  by  the  amount  5,,  assuming  the  direction  indicated  as  correct. 

Having  found  the  stress  '\n  AC  v^e  may  pass  to  the  joint  C.  The  unknowns  are  the 
stresses  in  CD  and  CB.  In  Fig.  25  {c)  the  joint  is  shown  free  with  all  forces  put  in;  5, 
will  point  towards  C,  AC  being  now  known  to  be  in  compression.  Assume  either  direction 
for  5,  and  S,.     As  before,  2  vert.  comp.  =  o  gives      cos  B  ^  —  W  —  S^=^  o  and 

2  hor.  comp.  —  o  gives      sin  ft  —  S,  sin  B  =  o,  from  which      and  5,  may  be  found. 

We  may  then  pass  to  D  or  B,  there  being  one  unknown  at  D  and  two  at  B,  one  of  which 
is  the  abutment  reaction. 


The  stresses  in  all  the  members  are  thus  found  by  treating  single  joints  in  succession, 
always  dealing  with  joints  at  which  not  more  than  two  unknown  forces  are  acting. 


API'LTCATION  OF  THE  LAWS  OF  EQUILIBRIUM. 


19 


Fig.  28. 


Example  2.  Let  it  be  required  to  find  the  stresses  in  all  the  members  of  the  truss  in 
Fig.  2(>;  P—  5000  lbs.,  /  =  50  ft.,  CG  —  10  ft. 

If  the  foregoing  problem  has  been  carefully  followed,  the  student  will  have  no  difficulty 
in  solvin-g  this  or  any  similar  problem. 

In  example  2  we  may  find,  if  desired,  the  stresses  in  BF  and  DH  at  once,  thus:  Fig. 
27  {a)  represents  the  joint  B  free;  5, ,  and  5,  are  the  stresses  in  the  members  cut. 
^  vert.  comp.  =  o  gives  at  once  S^  —  P  =  O,  or  S,  =  P\  likewise  in  Fig.  {b),  5,  =  P. 

In  general  we  see  that  if  two  of  three  unknown  forces  have  the  same  line  of  action,  the 
third  may  always  be  determined  by  putting  .2  comp.  perpendicular  to  this  line  of  action  =  o. 
This  principle  is  a  very  useful  one.  In  the  example  above,  after  finding  the  stress  in  BF,  we 
can  find  that  in  FC  by  equating  components  perpendicular  \.o  AG  at  joint  7^=0.  In  like 
manner  CH  can  be  found,  and  finally  CG. 

Example  3.  Find  the  stresses  in  all  the  members  of  the  first  two 
panels  in  Fig.  20,  Art.  31. 

Example  4.  Roof-truss  (Fig.  28).  P  —  2000  lbs.,  0  =  45°,  <<  =  60°, 
/  =  60  ft.    Find  the  stresses  in  all  the  members. 

Note. — Begin  at  D  and  find  stress  in  DF,  then  pass  to  F,  A,  etc. 

33.  Third.  Method  of  Sections. — In  this  method,  in- 
-stead  of  taking  single  joints,  a  section  is  passed  through  the 
structure  cutting  the  members  whose  stresses  are  desired,  and 
the  equations  of  equilibrium  applied  to  one  of  the  portions 
into  which  the  structure  is  thus  divided.  A  part  of  the  forces 
are  thus  external  forces,  and  a  part  are  due  to  stresses  in  the 
members  cut.  The  portion  of  the  structure  considered  usually  includes  several  joints,  and 
thus  the  forces  are  in  general  non-concurrent.  This  gives  us  three  equations  of  equilibrium, 
and  hence  if  we  cut  but  three  members  (not  meeting  in  a  point)  whose  stresses  are  unknown, 
these  stresses  may  be  found. 

Another  way  of  stating  the  equilibrium  existing  between  the  forces  acting  upon  either 
portion  of  the  structure,  is  to  say  that  the  stresses  in  any  section  hold  in  equilibrium  the 
e-xternal  forces  acting  upon  either  side  of  that  section.    Or,  more  in  detail, 

^  vert.  comp.  external  forces  =  5"  vert.  comp.  internal  forces; 
.2  hor.  comp.  external  forces  =  hor.  comp.  internal  forces  ; 
.5"  mom.  external  forces  =  2"  mom.  internal  forces. 

The  equality  is,  however,  one  of  numerical  value  but  not  of  sign,  as  is  seen  by  comparison 
with  the  fundamental  equations  of  equilibrium. 

/    >l  Example  1  (.same  truss  as  in   Fig.  26).  Re- 

quired the  stresses  in  BC,  FC.  and  FG.  We  fir.st  find 
the  abutment  reaction,  R \  by  methods  alread)-  given. 
Passing  a  section  through  tiie  above  members,  separat- 
ing the  portion  to  the  left,  and  replacing  the  stresses  in 
the  members  cut  bj' the  forces  5,,  .S".,,  and  ,S~3,  Fig  30, 
we  may  now  apply  our  three  equations.  ^  vert.  comp. 
=  o  gives  R' —  P-\-S„  sin  ^—  .Sj  sin  6^  =  0;  ^  hor. 
cos  0  —  S^=o\  mom.  about  F  =  o 
These  three  equations  enable  us  to  find  a' 


1 
1 

!  B 

P  \ 

■    1       c . 

p 

D, 

P 

F 

H  ' 

Fig.  29. 


comp.  =  o  gives  6,  cos  6 -\- S 
gives  R'  X  AB-  S^X  BF=  o. 
the  unknown  stresses.    The  student  may  substitute  numerical  values. 

R'  X  AB 

Notice  that  the  last  equation  gives  at  once  5,  =  — — ,  the  centre 
of  moment'5  being  at  F,  the  intersection  of  -S',,  and  .S",.     Similarlj-,  an 


4 


Fig.  30. 


equation  of  moments  with  centre  at  C  will  give  63  directly,  and  one  with  centre  at  A  will  give 


20 


MODERN  FRAMED  STRUCTURES. 


( 

IP 

f  ok 

/       \  >*<5 

<  

-   >^ 

,R'  R 

Fig.  31. 

;  thus  we  may  make  use  of  three  equations  of  moments.  The  lever-arms  of  the  forces 
when  not  directly  known  can  easily  be  computed  from  the  given  dimensions. 

Example  2.  Fig.  31  (same  truss  as  in  Fig.  28).    Find  the  stresses  in  all  the  members 

by  the  method  of  sections. 

We  first  get  R'  by  treating  the  structure  as  a  whole. 
Then  passing  a  section  mn,  cutting  but  three  members  not 
meeting  in  a  point,  we  separate  the  left-hand  portion,  put 
in  all  forces,  and  proceed  to  apply  our  three  equations, 
f      Fig.  32  {a). 

'  We  may  use  three  moment  equations,  taking  for  centres 

of  moments  the  points  yi,  iT,  and  C  successively  ;  or,  as  some 
of  the  lever-arms  are  tedious  to  compute,  a  better  method 
would  be  to  compute       by  a  moment  equation,  centre 
moments  at  C,  then  use  the  other  equations,  the  angles  d  and  a 
being  given.    In  any  case  that  equation  should  be  used  which 
gives  the  result  sought  in  the  simplest  way. 

Now  5,,  S^,  and  being  known,  a  section  op  through  DC, 
DF,  AF,  and  AB  cuts  but  two  pieces  whose  stresses  are  un- 
known. Fig.  {b).  Moment  equations  with  D  and  A  as  centers 
then  give  5,  and  S^. 

The  stresses  in  AF  and  AB  having  been  found,  the  stress 
in  AD  is  readily  found  by  passing  a  section  through  these  three 
pieces,  or  what  is  the  same  thing,  treating  the  single  joint  A.  The  stresses  in  the  remaining 
members  are  found  by  methods  similar  to  the  preceding. 

It  is  often  expedient  to  combine  the  method  of  sections  with  the  preceding  method  ; 
thus  in  the  present  example  the  stress  in  AB  may  be  found  by  passing  the  section  iim^  after 
which  we  may  pass  to  the  joint  A,  and  thence  by  single  joints.  A  single  application  of  the 
method  of  sections  to  this  problem  to  get  the  stress  in  AB  thus  enables  us  to  apply  the  oti^er 
method  in  a  regular  way,  beginning  at  A  and  passing  to  other  joints  in  turn. 


p 


Fig.  33.  Fig.  34. 


Example  3. 
Example  4. 
sections. 


Roof-truss  (Fig.  33)  P  =  3000  lbs.  Find  R'  and  R",  and  the  stresses  in  all  the  members. 
Bridge-truss  (Fig.  34)  F  =  5000  lbs.    Find  stresses  in  all  the  members  by  the  method  of 


Application  of  the  laws  of  equilibmivm. 


2i 


II.    Graphiral  Application. 
The   Equilibrium  Polygon. 


34.  Conditions  of  Equilibrium  of  Non-concurrent  Forces  Restated. — Reproducing 

Fig.  16  (see  Fig.  35),  ^^^3  was  found  to  be  the  resultant  of 
P^,  P^,  P3 ,  and  P^ ;  and  the  force  R^' ,  such  a  force  as  would 
balance  these  four  forces,  thus  forming  a  system  in  equilib- 
rium. The  conditions  of  equilibrium  were  found  to  be,  first, 
that  the  force  polygon  must  close  ;  second,  that  the  posi- 
tion of  the  last  force  (^3')  must  coincide  with  the  last  seg- 
ment {CD)  of  the  equilibrium  polygon. 

It  was  also  noted  that  each  segment  of  the  equilibrium 
polygon  is  the  line  of  action  of  the  resultant  of  all  the 
forces  upon  either  side  of  that  segment.  This  resultant  is 
given  in  amount  by  that  ray  in  the  force  polygon  to  which 
this  segment  is  parallel.  Its  direction  is  from  O  when  it  is 
the  resultant  of  the  forces  on  the  left  of  the  segment,  and 
toivards  O  when  it  is  the  resultant  of  the  forces  to  the  right. 

35.  Resolution  of  the  Forces. — Since  R^  is  the  result- 
ant of  Pj  and  Pj,  the  sum  of  the  horizontal  components  of 
P,  and  Pj  is  evidently  equal  to  the  horizontal  projection  of  Fig.  35. 

.  and  the  sum  of  their  vertical  components  is  equal  to  the 
vertical  projection  of  P,.    Similarly,  P,  with  respect  to  P, ,  P,,  and  P,;  and  in  general,  the  sjini 
of  the  horizontal  components  and  the  sum  of  the  vertical  components  of  all  the  forces  to  the  left 
of  any  segment  are  equal  respectiziely  to  the  horizontal  and  the  vertical  projections  of  the  ray  m 
the  force  polygon  to  zvhich  this  segment  is  parallel. 

36.  Moments  of  the  Forces. — It  follows  from  what  has  already  been  shown,  that  the 
sum  of  the  moments  of  all  the  forces  to  the  left  of  any  segment,  as  BC,  about  any  point  a,  is 
equal  to  the  product  of  the  parallel  ray  in  the  force  polygon,  (their  resultant),  multiplied 
by  the  perpendicular  distance,  ab,  from  the  point  to  the  segment  (the  line  of  action  of  the  re- 
sultant). Thus  the  sum  of  the  moments  of  P,  and  P,  about  a  —  R^  X  ac\  of  P,,  P,,  P,,  and 
P,  about  a  —  R^  X  ad,  etc. 

Let  ac'  be  drawn  vertically  through  a ;  also  project  P,  and  P,  upon  a  horizontal  line. 
Then  from  the  similar  triangles  abb'  and  <933',  we  have:  X  ab  =  X  nb' ,  or  the  sum  of 
the  moments  of  P, ,  P^,  andPj  about  a  =  Ol'  X  ab'  ;  and  similarly  the  sum  of  the  moments  of 
P,  and  P,  about  a  =  O2'  X  ac'.  We  have  then  this  useful  principle,  that  the  sum  of  the 
moments  of  the  forces  to  the  left  of  any  segment  about  any  point  is  equal  to  the  vertical  ordinate 
from  tlie  point  to  the  segment,  multiplied  by  the  horizontal  projection  of  the  corresponding  ray  in 
the  force  polygon.  It  is  evident  that  instead  of  vertical  ordinates  and  horizontal  projections 
we  may  use  any  two  directions  at  right  angles. 

The  foregoing  two  articles  apply  equally  well  to  the  forces  on  the  right  of  any  segment 
if  the  system  is  a  balanced  one. 

The  sign  of  the  moment,  or  the  direction  in  which  it  tends  to  turn  about  the  point,  is  at 
once  seen  when  we  remember  that  the  segment  is  the  line  of  action  of  the  resultant  of  the 
forces  in  question.  Thus  the  moment  of  P,  and  P,  about  a  is  right-handed  or  positive,  that  of 
P,,  Pj,  P3,  and  P,  is  positive,  that  of  R^  and  P,  is  negative,  etc. 


MODERN  FRAMED  STRUCTURES. 


Fig.  36. 


37.  Forces  Parallel  or  Nearly  so. — Given  the  forces      ,  and      (Fig.  36);  re- 
quired their  resultant. 

The  force  polygon  {p)  gives  their  resultant,  1-5, 
in  amount  and  direction  ;  5-1  is  a  force  which  will 
balance  the  given  system.  Resolve  this  force  into  any 
two  components,  P"  and  P\  by  drawing  the  triangle 
^0\,  O  being  chosen  so  as  to  give  fair  angles  at  i  and 
5.  These  components  together  with  the  given  system 
will  form  a  system  in  equilibrium,  provided  they  are 
inserted  in  Fig.  («)  in  the  proper  position.  The  posi- 
tion of  one  of  them,  as  P' ,  may  be  chosen  at  will,  and 
the  other  will  be  given  in  position  by  the  last  seg- 
ment of  the  equilibrium  polygon  constructed  for  the 
forces/",  .  .  .  P^,  with  O  as  the  pole  of  the  force 
polygon.  The  last  force  thus  coinciding  with  the 
last  segment,  and  our  force  polygon  closing,  equilib- 
rium is  assured.  The  intersection,  A,  of  P'  and  P"  is 
a  point  on  the  resultant  of  these  two  forces,  and 
since  5-1  gives  this  resultant  in  amount  and  direction,  it  is  therefore  fully  known.  It  is 
represented  by  R'  in  Fig.  {a),  and  since  it  balances  P^  .  .  .  P^,  an  equal  and  opposite  force, 
P,  is  the  required  resultant  of  our  given  system  in  amount,  direction  and  position. 

Corollary. — Since  the  resultant  of  all  the 
forces  up  to  any  segment  applied  along  that 
segment  would  balance  the  remaining  forces, 
it  follows  that  the  intersection  of  any  two  seg- 
ments of  an  equilibrium  polygon  is  a  point  on 
the  resultant  of  the  intermediate  forces.  Thus 
the  point  .5  is  a  point  on  the  resultant  of  /*, 
and  P^ ;  the  line  1-3  in  the  force  polygon  gives 
the  amount  and  direction  of  this  resultant. 

38.  Abutment  Reactions. — When  the 
given  system  of  forces  acts  upon  a  beam  or  frame- 
work supported  at  two  points,  the  abutment 
reactions  are  themselves  components  of  R' ,  Fig. 
36,  and  it  is  these  two  components  that  are 
usually  desired.  One  point  in  each,  the  abut- 
ment, is  given,  and  also  usually  the  direction  of 
one  ;  the  direction  of  the  other  and  the  magni- 
tudes of  both,  follow  from  the  conditions  of 
equilibrium.  It  will  now  be  shown  how  these 
reactions  may  be  fully  determined. 

Let  the  forces  P,  .  .  .  act  upon  any  rigid 
structure  ACB,  Fig.  37,  resting  upon  abut- 
ments at  A  and  B  (not  shown  in  the  figure). 
Suppose  the  end  B  to  rest  upon  rollers  and  the 
end  A  to  be  fixed  ;  the  reaction  at  B  will  be 
vertical,  that  at  A  unknown  in  direction. 

Construct  the  force  polygon,  Fig.  {b),  as  in  the  preceding  article,  choosing  any  point  O 
:is  pole.  Draw  the  corresponding  equilibrium  polygon,  AabcdB',  inserting  the  force  P'  at 
A ;  B'  is  the  intersection  of  P"  with  the  vertical  reaction  at  B.    Draw  the  line  AB' 


y.t 


H  W 

r  \ 

1 

/ 

Fig.  37. 


APPfJCATTON  OF  THE  LAWS  OF  EQUILIBRIUM. 


23 


called  the  closing  line  of  the  equilibrium  polygon,  and  On  parallel  to  it,  meeting  a  vertical 
through  5,  at  n.    Join  We  have  now  resolved  P"  into  the  components  5;/  and  nO,  and 

P'  into  On  and  n\.  Replacing  P"  and  P'  by  their  components,  inserted  at  ^' and  A  respec- 
tively, we  have  the  opposite  and  equal  forces  N'  and  N"  acting  along  the  same  line  AB'\ 
hence  they  balance  each  other  and  the  forces  T'  and  T"  must  alone  hold  P^  .  .  .  in  equi- 
librium. They  are  therefore  the  required  reactions,  as  they  fulfil  all  conditions.  That  the\- 
are  components  of  5-1,  or  R' ,  is  seen  from  the  force  polygon. 

If  the  direction  of  T"  had  been  unknown  and  that  of  T'  known,  then  we  should  have 
begun  our  equilibrium  polygon  at  B  instead  of  at  /Z,  and  worked  towards  the  left. 

If  both  ends  of  the  structure  are  fixed,  T'  and  T"  are  parallel,  which  can  be  true  only 
when  they  are  both  parallel  to  R' .  In  that  case  we  know  the  direction  of  both  reactions  and 
our  equilibrium  polygon  need  not  pass  through  either  or  B  as  the  extremities  of  the  closing 
line  are  then  the  intersections  of  P'  and  P"  with  lines  through  A  and  B  parallel  to  R' .  In  the 
above  problem,  under  those  conditions,  AB"  v^o\x\<\  be  the  closing  line,  and  5;/  and  n  \  the 
abutment  reactions,  BB"  being  parallel  to  5-1,  and  On'  parallel  to  AB" . 

39.  Resolution  of  the  Forces. — In  Fig.  37,  the  equal  and  opposite  forces  N'  and  N"  do 
not  really  act  upon  the  structure,  they  being  virtually  inserted  at  the  time  we  assume  a  con- 
venient pole.  They  do  not,  therefore,  enter  among  the  external  forces  in  finding  stresses  in 
the  structure.  It  must  be  borne  in  mind,  however,  that  these  forces  are  always  included  when 
we  speak  of  the  segments  of  the  equilibrium  polygon  as  being  the  lines  of  action  of  certain 
resultants. 

The  sum  of  the  vertical  and  the  sum  of  the  horizontal  components  of  all  forces,  actually 
acting  upon  the  structure  to  the  left  of  any  segment,  as  be,  are  evidently  equal  to  the  vertical 
and  the  horizontal  projections,  respectively,  of  their  resultant,  «3.  If  the  force  N'  be  included, 
the  resultant  of  all  forces  to  the  left  of  be  is  6^3,  the  vertical  and  horizontal  projections  of  which 
are  equal  to  the  sums  of  the  corresponding  components  of  the  forces. 

40.  Moments  of  the  Forces  (Fig.  37). — The  sum  of  the  moments  of  N' ,  T',  and  P^ 
about  any  point  x  is,  as  in  Art.  36,  equal  to  the  vertical  ordinate  xy  multiplied  by  O2',  O2' 
being  the  horizontal  component  of  the  resultant,  O2,  of  the  three  forces. 

If  now  we  wish  to  eliminate  the  moment  of  the  imaginary  force  N',  we  can  take  our  cen- 
tre of  moments  upon  AB' ,  its  line  of  action,  thereby  reducing  its  moment  to  zero.  Thus  the 
sum  of  the  moments  of  T'  and  P^  about  x^  =  x^yx  O2' ;  the  sum  of  their  mornents 


'(bout  =  x^y^  X  O2',  etc.  To  get  their  moments  about  x,  we  draw  xx'  parallel  to  their 
-►■esultant,  «2.  The  sum  of  their  moments  about  all  points  in  xx'  is  the  same,  hence  their 
moment  about  x  =  x'j'  x  O2'.  The  sign  of  this  moment  is  deteimined  by  applying  the  force 
O2  along  ab ;  as  this  force  acts  right-handed  about  x',  the  above  moment  is  positive. 


24 


MODERN  FRAMED  STRUCTURES. 


41.  Application  to  a  Beam. — Let  it  be  required  to  find  the  abutment  reactions,  T'  and 
T" ,  of  tlie  beam  AB,  Fig.  38,  supporting  tlie  loads  P^.  Beam  fixed  at  eacli  end  against 
horizontal  motion. 

Fig.  {b)  shows  the  force  polygon  ;  i  2,  ...  6  being  the  load  line,  and  O'  the  pole.  The 
reactions  will  be  parallel  to  6-1  ;  A'  and  are  their  intersections  with  /"and  P" .  We  get 
for  their  values,  T'  and  T"  —611.    The  student  may  follow  out  the  details  of  the 

construction. 

If  we  wish  the  sum  of  the  moments  of  the  real  forces  to  the  left  of  any  section,  mn, 
about  the  neutral  axis,  x,  of  the  section*,  we  may,  as  in  Art.  40,  draw  xx'  parallel  to  the 
resultant  ;/3  of  these  forces  {T',      ,  and  /\),  whence  the  required  moment  =  x'y'  X  O't,'. 

Where  many  moments  are  required  it  will  be  more  convenient  to  draw  a  new  equilibrium 
polygon  whose  closing  line  shall  pass  through  the  centres  of  moments,  or  coincide  with  the 
neutral  axis  of  the  beam.  This  requires  a  new  pole  for  our  force  polygon,  so  chosen  that  the 
equilibrium  polygon,  if  made  to  pass  through  A,  will  also  pass  through  B.  The  abutment 
reactions  depending  only  upon  the  given  system  of  forces  and  the  positions  of  A  and  B,  the 
point  71  in  our  force  polygon  will  not  be  changed;  and  as  the  closing  line  is  to  be  AB,  the 
required  pole  must  lie  somewhere  on  the  line  iiO  drawn  parallel  to  AB.  With  any  point  O 
on  this  line  as  pole,  draw  the  force  polygon,  and  beginning  at  A  construct  the  corresponding 
equilibrium  polygon  ;  it  will  pass  through  B  if  accurately  drawn.  The  vertical  ordinate  from 
the  neutral  axis  of  the  beam  to  the  polygon,  multiplied  by  the  horizontal  projection  of  the 
proper  ray  in  the  force  polygon,  then  gives  the  sum  of  the  moments  of  all  the  forces  actually 
acting  upon  the  beam  to  the  left  or  right  of  the  centre  of  moments.  This  moment  is  positive 
for  the  forces  on  the  left  and  negative  for  those  on  the  right. 

42.  Problem.    To  Pass  an  Equilibrium  Polygon  through  Three  Given  Points. — 

Let  P,,  P^,  and  P^  be  the  given  forces  (Fig.  39)  and 
A,  B,  and  C  the  given  points. 

Draw  a  force  polygon,  (d),  with  any  pole  O',  and 
the  corresponding  equilibrium  polygon,  AabcB' ,  pass- 
ing through  the  point  A,  the  reaction  line,  BB', 
being  drawn  parallel  to  4-1.  Draw  the  closing 
line  AB'  and  the  parallel  line  O  n  ;  4«  and  nl  would 
be  the  abutment  reactions  were  the  given  forces 
acting  upon  a  beam  with  fixed  ends,  abutments  at  A 
and  B.  By  the  preceding  article,  the  required  pole 
lies  somewhere  on  the  line  71O,  drawn  parallel  to  AB. 

In  like  manner  treat  the  two  forces  P^  and  P^ 
and  the  points  A  and  C.  The  trial  force  polygon, 
O'  I  2  I  and  the  equilibrium  polygon  Aabc  are 
already  drawn.  The  resultant  of  P^  and  P^  is  1-3, 
and  a  line  through  C  parallel  to  1-3  intersects  be  at 
C.  The  closing  line  is  AC,  and  the  line  O'ti'  drawn 
parallel  to  it  meeting  1-3  at  71'  determines  the  reac- 
tions at  A  and  C  for  loads  P,  and  P^.  In  order  that 
the  required  equilibrium  polygon  may  pass  through 
C,  the  pole  of  the  force  pol}'gon  must  lie  somewhere 
on  the  line  71' O  drawn  parallel  to  AC.  Hence  the 
required  pole  O  lies  at  the  intersection  of  71' O  and  71O. 


Fig.  39. 


*The  student  is  assumed  to  be  familiar  with  the  -theory  of  flexure.  If  he  has  not  studied  this  subject,  he  is 
referred  to  Chap.  VIII. 


APPLICATION  OF  THE  LAWS  OF  EQUILIBRIUM. 


The  load 


43.  Parallel  Forces. — Let  P,,  P^,  and  Fig.  40,  be  a  system  of  parallel,  and  in  this  case 
vertical,  forces  acting  upon  any  structure  with  abutments  A  and  B. 

The  abutment  reactions,  which  are  both  vertical,  are  found  in  the  usual  way. 
line  is  the  vertical  line  1-4.    The  reactions  T"  and  T' 

—  4n  and  ni  respectively. 

The  sum  of  the  vertical  components  of  the  forces,  or  of 
the  forces  themselves,  actually  acting  upon  the  structure 
to  the  left  of  any  segment,  is  equal  to  the  distance  from  n 
to  the  extremity  of  the  ray  parallel  to  the  segment.  Thus 
the  sum  of  T'  and      =  «2  ;  of  T',  P^,  and  P^  =  n^,  etc. 

The  sum  of  the  horizontal  components  =:  zero. 

The  sum  of  the  moments  of  T'  and  about  any 
point  X  =,as  before,  x'j  X  Om,  where  ;ir.ar'  is  drawn  parallel 
to  the  resultant  of  T'  and  P,  (vertical  in  this  case),  and 
Om  is  the  horizontal  projection  of  O2.  Likewise  the  sum 
of  the  moments  of  T',Pj,  and  P^  about  x^  =  x/j^  X  Om,  etc.  The  distance  Om  is  here 
called  the  pole  distance  o{  the  force  polygon.  We  have  then  in  general:  For  vertical  forces, 
t/ie  Sinn  of  the  moments  of  all  the  external  forces  to  the  left,  or  right,  of  any  segment,  about  any 
point,  is  equal  to  the  intercept  on  the  vertical  ordinate  tJirough  the  point  included  between  the 
closing  line  and  that  segment,  multiplied  by  t lie  pole  distance  of  tlie  force  polygon. 

If  x'y  X  Om  —  sum  of  the  moments  of  T'  and  /*,  about  x,  and  x'y'  X  Om  —  sum  of  the 
moments  of  T' ,  and  P^  about  the  same  point,  then  must  yy'  X  Ovi  =  moment  of  P„  about 
x.  Hhdit  IS,  the  momeJit  of  any  force  about  any  point  is  equal  to  the  intercept  on  the  vertical 
ordinate  through  the  point  included  betzveen  the  adjacent  segments  of  the  equilibrium  polygon, 
multiplied  by  the  pole  distance.    This  is  known  as  Culmanii  s  Principle. 

The  above  discussion  applies  equally  well  to  parallel  forces  in  any  direction,  provided  the 
abutment  reactions  are  also  in  the  same  direction.  When  that  is  not  the  case,  as  when 
inclined  forces  act  upon  a  structure  where  one  end  reaction  is  vertical,  we  must  follow  the 
general  method  of  Arts.  38-40. 

44.  Forces  Taken  in  Any  Order. — Suppose  the  forces  P,,  P,,  P^,  and  /*„  Fig.  41,  to 
act  upon  the  beam  supported  at  A  and  B. 

Lay  off  the  load  line  1-5,  taking,  for  variety,  the  forces  in  the  order  and  P^. 

Draw  the  force  polygon  with  pole  O  and  the  correspond- 
ing equilibrium  polygon,  beginning  at  a  point  in  the 
vertical  through  A.  The  equilibrium  polygon  is 
A'abcdB',  and  the  closing  line  A'B'.  The  line  On  drawn 
parallel  to  A'B'  determines  the  abutment  reactions,  T" 
and  T',  these  being  equal  respectively  to  5«  and  ti\  both 
in  amount  and  direction  ;  T'  therefore  acts  downwards. 

The  sum  of  the  moments  of  T'  and  P^  about  x 

—  x'y  X  Om  and  is  negative.  The  sum  of  the  moments 
of  T',  P,,  and  P^  about  a  point  x,  is  not  given  by  a  single 
ordinate  rnultiplied  by  Om,  since  the  force  P^  does  not 
come  next  in  the  construction.  The  sum  of  the  moments 
of  T'  and  P^  about  x,  =  x/y^  x  Om,  and  the  moment  of 
P,  =  zz'  X  Om  ;  hence,  these  moments  being  of  the  same 
sign,  the  required  %\\m-={x'y^-\-zz')  X  Om.  Similarly, 
the  sum  of  the  moments  of  T' ,  P  ,        T" ,  and  P^  about 

—  {x^y^  —  x^z^  X  Om.    Thus  by  a  little  inspection  any  desired  moment  may  be  found. 


Fig.  41. 


26 


Modern  framed  structures. 


Application. 

45.  First.  To  the  Structure  as  a  Whole  to  determine  a  portion  of  the  external 
forces. 

Example  i.  Three  equal  cylinders,  each  weighing  1000  lbs.  (Fig.  42),  B  and  C  just 
touching  at  k ;  surfaces  smooth  so  that  pressures  are  normal.  Required  the 
reactions  at  c,  d,  e,  and  /and  the  pressures  at  g  and  h. 

Fig.  43  shows  each  cylinder  with  all  external  forces  indicated.    We  have 
here  three  systems  of  concurrent  forces  in  equilibrium,  hence  the  force  polygon 
^  d       c         of  each  system  must  close. 

iG.  42.  Fzrsi,  the  system  acting  upon  A,      and      being  unknown.    In  Fig.  44  {a) 

lay  off  0\  =  1000  lbs.,  to  scale  ;  then  draw  1-2  parallel  to  P^,  and  O2  parallel  to  P^,  cutting  1-2 
at  2  (/",  and  P,  will  be  inclined  30^  from  the  vertical).    Then  1-2  and  2O  are  the  amounts  of 
the  forces  necessary  to  close  the  polygon, 
when  acting  in  the  directions  given  by/!, and 


Ri 


Fig.  43. 


Fig.  44. 


P^.    Therefore  /*,  and  P^  are  respectively 
equal  to  2O  and  1-2. 

Second,  t/ie  syste7H  acting  upon  B.  The 
forces  Q,  and  R,  are  unknown.  In  Fig.  44 
{b)  draw  0\  equal  and  parallel  to  P^  as  found 
from  {a)  but  in  the  opposite  direction  ;  then 
1-2  vertically  and  equal  to  1000  lbs.  The 
lines  2-3  and  3O  drawn  parallel  to  and 
close  the  polygon,  and  give  these  forces  in 
amount.    They  are  thus  fully  determined. 

Third,  the  system  actittg  upon  C.    A  figure  {c)  similar  to  {b)  gjves      —  2-3  and      =  3(9. 
The  numerical  values  are  found  by  scale  to  be  as  follows:  P^—P^  —  580  lbs. ;  Q^  — 
=  270  lbs. ;  R^=  R^~  \  500  lbs. 

We  see  from  the  above  that  two  unknown  forces  in  any  concurrent  system  are  easily 
found  by  closing  the  force  polygon  by  lines  parallel  to  the  directions  of  these  two  forces. 

A  few  examples  will  now  be  given  as  a  further  illustration  of  the  application  of  the 

equilibrium  polygon  in  finding  abutment  reactions. 

Example  2  (same  truss  as  in  Fig.  18).  The  unknowns 
are  R\  R",  and  6. 

We  have  here  a  system  of  non-concurrent  forces  in  equi- 
librium, and  hence  the  two  conditions :  their  force  polygon 
must  close  and  the  last  force  must  coincide  with  the  last  seg- 
ment of  the  equilibrium  polygon.  Beginning  with  the  known 
forces,  W  diwd  G,  draw  the  corresponding  portion  of  the  force 
polygon,  O  I  2,  and  the  ray  O2  ;  also  the  segment  AB,  parallel 
to  this  ray.  The  direction  of  R'  being  known  we  take  this  force 
next  in  order;  B  is  its  intersection  with  AB.  The  amount  of 
R'  being  unknown,  we  cannot  at  once  draw  the  next  ray  in  the 
force  polygon  and  so  get  the  direction  of  the  next  segment  of 
the  equilibrium  polygon.  We  know,  however,  that  this  next 
segment  must  coincide  with  the  last  force  R",  and  hence  must 
pass  through  E ;  BE  is  therefore  this  segment.  The  ray  6^3  is 
then  drawn  parallel  to  BE  to  its  intersection  with  the  line 
2-3  drawn  parallel  to  R' .    The  figure  O  i  2  3  6*  is  a  closed 


Bis 


Fig.  45. 


polygon,  and  R'  and  R"  are  respectively  equal  to  2-3  and  3C?;  BE  is  the  line  of  action  of  R". 


APPLICATION  OF  THE  LAV/S  OF  EQUILIBRIUM. 


27 


Example  3.  Find  graphically  the  abutment  reactions  of  the  truss  in  Fig.  20,  a  case  of  parallel  forces. 

Example  4.  Given  the  roof-truss,  Fig.  46,  with  loads  .  .  .  and  wind-pressures 
W,  ,  IV,,  JVj.  Required  the  abutment  reactions, 
R'  and  R"  ■  first,  when  the  horizontal  thrust  due 
to  wind  is  taken  up  by  each  abutment,  the 
truss  being  fixed  at  both  ends ;  second,  when 
one  end  is  on  rollers,  the  other  end  taking  all  the 
thrust. 

Construct  the  force  polygon  (a),  laying  off  the 
forces  in  any  convenient  order.  The  order  chosen 
is  Pj  ,  P,,  P,,  W^,  W„,  etc. ;  1-9  is  their  resultant. 
Since  many  of  the  forces  are  parallel,  it  will  be 
necessary  to  resolve  9-1  into  the  components  P' 
and  P"  by  selecting  a  pole  O.  Draw  the  rays 
O2,  O3,  etc.,  and  construct  the  corresponding 
equilibrium  polygon,  beginning  at  any  point  A', 
on  a  line  through  A  parallel  to  1-9.  In  construct- 
ing this  polygon  we  must  be  careful  to  take  the 
forces  in  the  same  order  in  which  they  occur  in 
the  load  line.  The  polygon  is  A'abcdefghB' ,  A'B' 
being  the  closing  line.  The  line  On  drawn  par- 
allel to  A'B'  gives  the  lequired  reactions  ;  R'  =  ni 
And  R"  —  9//. 

If,  for  example,  the  left  end  be  on  rollers,  the  only  part  of  R'  acting  upon  the  truss  at  A 
is  its  vertical  component,  =  Ni.  Its  horizontal  component  can  be  applied  only  at  B,  another 
point  in  the  line  of  action  of  this  component,  and  a  point  where  the  truss  is  fastened  to  the 
abutment.  This  horizontal  component,  uN,  combined  with  R" ,  or  gn,  gives  9iVas  the  result- 
ing reaction  at  B. 

For  this  last  case  the.  reactions  might  have  been  found  directly  by  following  the  method 
of  Art.  38  ;  that  is,  by  beginning  our  polygon  at  B,  the  fixed  end,  and  ending  it  in  a  vertical 
through  A.  The  closing  line  would  have  been  some  line  BA" ,  and  the  line  in  the  force  poly- 
gon parallel  to  it  would  be  ON,  giving  the  same  reactions  as  found  above. 

46.  Second.  To  Single  Joints  to  Find  Stresses:  Maxwell  Diagrams. — In  this  method 
we  first  find  the  abutment  reactions  either  analytically  or  graphically,  then,  commencing  at  one 

abutment,  find  the  stresses  in  the  members  at  successive  joints 
by  means  of  force  polygons  only.  We  must  always,  as  in  the 
analytical  method,  select  joints  at  which  there  are  but  two  un- 
known stresses;  or  if  there  are  three,  we  can  determine  one  if 
the  other  two  have  the  same  line  of  action. 

Example  i  (Fig.  47).  P^  and  P^  each  =  500  lbs. ;  = 
1000  lbs.  ;  dimensions  as  given.  Required  the  stresses  in  the 
members. 

We  see  at  once  that  R'  =  R"  =  1000  lbs.  For  joint  /  we 
lay  of?  in  {a),  Oi  =  lOOO  lbs.  by  scale,  and  parallel  to  R' ;  1-2 
=  500  lbs.  downwards  ;  and  2-3  and  O^  parallel  to  the  pieces 
/w  and  Ifi  respectively.  Then  2-3  =  stress  in  /;;/,  and,  pointing 
in  the  direction  w/,  it  indicates  compression  ;  ^O  —  tensile  stress 
in  /n.  Fig.  (d)  is  the  force  polygon  for  joint  w,  2-3  being  the 
compression  in  7/10  and       the  tension  in  The  other  joints  are  treated  similarly.  Scaling 

off  the  stresses  from  the  force  polygons  we  have  the  following  results :  stress  in  /w  =  stress 


Fig.  47. 


28 


MODERN  FRAMED  STRUCTURES. 


Fig.  48. 


\n  mo  =  1800  lbs.  compression.  Stress  in  =  stress  in  no  =  1580  lbs.  tension.  Stress  in 
mn  —  1000  lbs,  tension. 

Instead  of  drawing  a  separate  figure  for  each  joint,  we  may  combine  the  force  polygons 
into  a  single  figure,  using  each  line  twice  as  being  common  to  two  polygons.  The  combined 
figure  may  be  called  a  stress  diagram.  A  convenient  notation  to  accompany  this  method  is 
to  letter  each  triangle  of  the  truss,  and  also  each  space  between  the  external  forces,  as  in 
Fig.  47.  Each  piece  and  each  external  force  is  then  known  by  the  two  letters  in  the  adjacent 
spaces,  as  the  piece  CF,  the  load  CD,  etc. 

Let  us  apply  this  method  to  tlie  truss  in  the  figure.  We  will  find  it  convenient  to  lay  off 
the  loads  and  reactions  at  once,  forming  the  load  line  BE,  Fig.  48,  the  loads  being  lettered  to 
correspond  with  Fig.  47.  The  abutment  reactions  are  EA  and  AB.  (If  these  are  found 
graphically  by  means  of  an  equilibrium  polygon,  we  will  have  our  load  line 
already  laid  off.)  Beginning  as  before  at  joint  /,  the  force  polygon  for  that 
joint  will  be  ABCFA  ;  CF  the  stress  in  piece  CF,  and  FA  that  in  piece  FA. 
The  nature  of  these  stresses  is  easily  determined  by  following  around  the 
polygon.  Passing  to  joint  m,  the  force  polygon  will  be  FCDGF,  in  which  FC 
and  CD  are  already  drawn.  The  force  FC  of  course  acts  in  the  opposite  direc- 
tion from  what  it  did  upon  joint  /.  The  polygon  for  joint  n  \s  A  EG  A ,  a.x\d  {or  0 
is  AGDEA,  EA  being  the  abutment  reaction  :  its  coming  out  so  is  a  check  upon  the  work.  In 
treating  a  joint  always  begin  with  the  piece  farthest  to  the  left  whose  stress  is  known,  and  then 
pass  around  right-handed.  Thus,  at  joint  m  the  forces /^C  and  CD  diXQ  known;  begin  then 
with  FC  and  pass  to  D,  G,  and  F;  etc.  If  0  had  been  the  starting  point,  the  opposite  mode  of 
procedure  would  liave  been  the  most  convenient.  1 

Example  2.  Required  the  stresses  in  the  roof- 
truss  of  Fig.  49.  Loads  NO  and  VW  each  =  1000 
lbs.  ;  all  others  =  2000  lbs.  Span  =  100  ft., 
rise  =  35  ft. ;  distance  from  apex  to  horizontal  tie, 
XM,  —  30  ft. ;  all  struts,  AB,  CD,  EE,  etc.,  are 
normal  to  roof. 

The  load  line  is  NW\  abutment  reactions, 
WM  and  MN.  The  stress  polygons  for  the  first 
three  joints,  beginning  at  the  left,  are  readily  drawn. 
At  the  third  upper  joint  {BPQ,  etc.),  however,  the 
stresses  in  the  three  pieces  CD,  DE,  and  EQ  are 
unknown.  As  in  Art.  32,  Ex.  4,  we  may  pass  to  the 
next  upper  joint,  find  the  stress  in  EF,  then  in  ED, 
and  finally  the  stresses  in  CD  and  QE.  As  to  the 
stress  in  EF: — From  Q  and  R  draw  the  indefinite 
lines  QE'  and  RF'  parallel  to  the  pieces  QE  and 
RF.  The  stresses  in  the  pieces  QE  and  RF  are 
unknown,  but  whatever  they  are,  the  stress  in  EF 
must  be  such  as  to  close  the  force  polygon  E' QRF  when  drawn  parallel  to  the  piece  The 
line  EE'  then  gives  this  stress,  a  compressive  one.  To  get  the  stress  in  ED  we  may  in  a  similar 
manner  draw  the  indefinite  line  F' D'  parallel  to  piece  FX,  then  the  line  E' D'  parallel  to  piece 
ED ;  D' E'  will  be  the  required  tensile  stress.  Knowing  ED,  we  may  now  pass  to  the  third 
upper  joint.  The  portion  CBPQ  of  the  force  polygon  is  already  drawn.  Taking  the  piece 
ED  next  in  order,  its  stress  being  known,  draw  QD"  equal  and  parallel  to  E' D' ;  then  D" D 
parallel  to  QE,  meeting  CD  at  D.  We  have  then  D" D  equal  to  the  stress  in  QE,  and  DC 
that  in  piece  DC.  The  forces  may  now  be  arranged  in  proper  order,  Q,  E,  D,  C,  in  the 
force  polygon.    The  remaining  stresses  are  easily  found. 


Fig.  49. 


APPLICATION  OF  THE  LAWS  OF  EQUILIBRIUM. 


29 


Fig.  50. 


An  easier  solution  of  the  above  example  is  now  seen  from  the  diagram.  It  depends  upon 
the  fact  that  E,  F,  A  and  B  are  in  the  same  straight  line, 
and  to  find  E  and  F  we  have  only  to  produce  AB  to  cut 
the  lines  QE  and  RF.  We  may  then  draw  FX  parallel  to 
the  piece  FX,  and  CD  likewise,  thus  determining  the 
point  D.  The  half  diagram  is  completed  by  drawing  XM 
and  DE.  This  method  is  applicable  only  when  the  struts 
are  normal  to  the  roof,  and  when  the  secondary  truss  NS 
is  symmetrical  about  CD. 

To  aid  in  distinguishing  the  nature  of  the  stresses, 
lines  representing  compressive  stresses  may  be  made  heavier 
than  those  representing  tensile  stresses,  or  different-colored 
inks  may  be  used.  Where  the  loads  are  symmetrical  it  is 
necessary  to  draw  but  one  half  the  stress  diagram,  the 
stresses  in  corresponding  members  of  the  two  halves  of  the 
structure  being  equal.  Where  this  is  done,  however,  the 
work  should  be  checked  by  computing  a  few  of  the  stresses 
by  analytical  methods. 

Fig.  50  shows  a  truss  with  accompanying  stress  dia- 
gram.   A  portion  of  the  loads  are  inclined,  due  to  wind- 
pressure  from  the  right.    Rollers  under  right  end  ;         =  right  abutment  reaction,  iV(9  the 
left. 

Example  3.  Find  all  stresses  in  the  bridge-truss  shown  in 
Fig.  51.    Each  load  =  30,000  lbs. 

47.  Special  Application. — In  case  we  desire  the 
stresses  in  but  one  or  two  members  of  the  structure 
when  under  a  given  loading,  the  preceding  method 
necessitates  the  finding  of  all  the  stresses  up  to  the  ones 
in  question,  and  thus  is  a  much  longer  process  than  the 
analytical  method  of  sections.  Where  there  are  no  loads  between  one  of  the  abutments  and 
these  members,  as  is  frequently  the  case,  we  may  find  the  desired  stresses  very  quickly  by 
graphics  as  follows : 

Suppose  we  wish  to  find  the  stresses  in  the  pieces  CD,  DE,  and  EF  (Fig.  51),  there 
being  no  loads  to  the  left  of  a. 

Pass  a  section  through  these  pieces.  Fig.  52  shows  the  left-hand 
portion  free,  with  the  forces  put  in;  .S, ,  >  ^""^  ^3  are  the  unknown 
stresses.  (The  load  P  is  not  at  present  supposed  to  exist.)  Now 
these  stresses  with  R'  form  a  balanced  system  whose  equilibrium 
is  independent  of  the  shape  of  the  structure  acted  upon.  We  may 
therefore  replace  the  portion  Accd  by  a  single  triangle,  Acd,  the 
structure  still  being  a  rigid  one.  If  we  then  begin  at  A  and  find 
the  stresses  in  the  members  of  this  new  structure,  the  force  polygons 

for  the  joints  c  and  d  will  give  us  the  required  stresses.    The  complete  diagram  is  given  in 
^  Fig.  53.    Beginning  at  F,  FC  is  the  abutment  reaction  R'  (R'  is  best  found 

analytically) ;  FCGF,  the  force  polygon  for  joint  A  ;  FGEF,  that  for  joint  d, 
there  being  no  stress  in  piece  cd,  as  it  exists  in  our  new  structure  ;  EGCDE 
is  the  polygon  for  joint  c.  We  have  then  EF  =  S,,  CD  =  5, ,  and  DE  =  5,. 
If  we  desire  the  stress  in  cd  as  a  part  of  the  original  truss,  we  may  pass  to 
joint  c  of  that  truss.  There  are  but  two  unknown  stresses.  The  force  polygon  EDCG'E 
gives  these  stresses  ;  G'E  =  stress  in  cd. 


Fig.  52. 


S3 

Fig.  53. 


MODERN  FRAMED  STRUCTURES. 


Fig.  54, 


If  there  be  also  a  load  at  d,  our  triangle,  Acd,  will  still  give  us  a  rigid 
structure.  The  stress  diagram  will  then  be  as  in  Fig,  54,  where  F' F  = 
load  P. 

Let  the  student  find  the  numerical  values  of  the  stresses  in  botn  cases, 
each  load  being  equal  to  30,000  lbs.,  and  compare  with  those  found  in 
Ex.  3,  above. 

48.  Third.  To  the  Method  of  Sections. — A.  To  Successive  Sections,  commencing  at 
One  End.—\{  we  pass  a  section  through  a  structure,  cutting  but  two  members  whose  stresses 
are  unknown,  the  single  condition  that  the  force  polygon,  drawn  for  the  forces  acting  upon 
one  portion  of  the  structure,  must  close,  will  enable  us  to  find  the  stresses  in  these  pieces. 
Commencing  at  one  end  of  a  structure  and  passing  a  section  cutting  but  two  pieces,  we  can 
determine  their  stresses ;  then  passing  another  section  cutting  three  members,  one  of  which 
has  already  been  treated,  we  can  find  the  stresses  in  the  other  two,  and  finally  by  successive 
sections  we  can  determine  all  the  stresses  by  simple  force  polygons. 

Take,  for  example,  the  truss  in  Fig.  55.    Lay  off  at  once  the  load  line,  MR  (Fig.  56) ;  RA 

and  ^J/are  the  reactions  at  right  and  left  abutments,  found  by 
any  method.    Pass  a  section  through  ABM,  cutting  two  pieces. 
A  simple  force  polygon,  AMB  (Fig.  56),  gives  the  stresses  in  MB 
and  BA. 

(In  this  case  it  is  more  convenient  to  work 
left-handed,  the  forces  in  the  load  line  being 
laid  off  in  left-handed  order  in  the  sense  of 
rotation  about  the  truss.) 

Now  pass  a  section  through  ABCN.  The 
stresses  in  BC  and  CN  are  unknown.  Fig. 
55  {d)  shows  the  portion  to  the  left  of  the  sec- 
tion free.  The  forces  acting  are  AM,  MN, 
NC,  CB,  and  BA,  and  since  they  are  in  equilibrium  their  force  polygon  must  close.  The 
portion  BAMN  {^\<g.  56)  of  the  polygon  is  already  drawn  ;  and  CB  drawn  parallel  to  their 
respective  pieces  closes  the  polygon  and  determines  the  stresses  in  these  pieces.  Next  pass  a 
section  through  ADCN;  CD  and  DA  are  the  unknown  stresses.  Fig.  {b)  shows  the  portion 
of  the  structure  considered.  Of  the  force  polygon  for  the  forces  there  acting,  the  portion 
AMNC  is  drawn.  The  lines  CD  and  DA  close  the  polygon  and  determine  the  unknown 
stresses.  In  like  manner  proceed  throughout  the  structure.  The  complete  stress  diagram  is 
given  in  Fig.  56. 

While  the  above  is  a  different  method  than  that  given  in  Art.  46,  yet  the  resulting  dia- 
gram is  precisely  the  same  as  would  have  been  obtained  b}'  that  method,  the  force  polygon 
for  any  joint  being  given  directly  in  the  figure.  Moreover,  if  we  pass  an\-  section  whatever 
through  the  structure,  the  polygon  of  the  forces  acting  upon  either  portion  will  be  given  by 
the  diagram.  Thus,  passing  a  section  the  forces  acting  upon  the  left-hand  portion  are 
the  loads,  abutment  reaction,  and  the  stresses  in  the  members  cut.  Their  force  polygon 
is  AMNOPHGFEDCBA,  a  closed  figure.    Likewi.se  with  the  portion  on  the  right. 

B.  To  any  Section  by  the  Method  of  Moments. — The  preceding  article  gives  a  special 
method  of  .applying  graphics  to  find  stresses  in  members  cut  by  certain  sections.  We  will 
now  show  how  the  equilibrium  polygon  may  be  applied  to  the  general  case. 

Take,  for  example,  the  roof-truss  in  Fig.  57.  Draw  the  force  polj'gon  0\  .  .  .  6,  and  the 
corresponding  equilibrium  polygon  A'bcdB'\  6n  and  ni  are  the  abutment  reactions,  R"  and 
R'.    Suppose  we  wish  to  find  the  stresses  in  the  pieces  cut  by  the  section  Im. 

Consider  the  portion  to  the  left  of  the  section  free;  the  forces  acting  are  R',P^,  , 
and  the  stresses  in  the  three  members.    Equating  the  sum  of  the  moments  about  C  equal 


B 

C 

\  E 

/ 

L 

Fig.  55. 


Fig.  56. 


APPLICATION  OF  THE  LAWS  OF  EQUILIBRIUM. 


31 


to  zero,  we  will  have  :  mom.  of  stress  in  AD  sum  of  the  moments  of  P^,  P, ,  and  =  o. 
The  second  term  of  this  equation  is  given  by  bb'  X  Ofn, 
hence  (stress  '\x\  AD)  FC  ^  bb' y^Om  —  o.  The  sign  of 
the  second  term,  or  of  the  moment  of  the  external  forces, 
is  seen  by  applying  the  force  (93  in  the  line  be.  It  acts 
right-handed  about  b' ,  and  as  its  moment  about  b'  is  the 
same  as  the  moment  of  the  forces  ,  and  R'  about  C, 
the  sign  of  our  second  term  is  positive.*  The  first  member 
is  therefore  negative  ;  that  is,  the  stress  in  AD  acts  left- 
handed  about  C.   AD  is  then  in  tension  by  the  amount 

— •    To  get  the  stress  in  CD  take  centre  of  moments 

at  A.    The  sum  of  the  moments  of  R' ,     ,  and  P^  about 

A  =  A'a"  X  Om  and  is  positive.     The  moment  of  the 

stress  in  CD  is  then  negative  or  left-handed  about  A.  The 

A'a'xOm      ^  .  .         ^  ,., 

■,  and  IS  compressive.     In  like  manner 

c'c  X  Om 


stress  - 


AG 


Fig.  57- 


the  compressive  stress  in  CE  = 


DH 


The  stresses  in  all  the  members  may  be  found  by  further  application  of  this  method. 
The  sections  must  always  be  so  chosen  as  not  to  cut  more  than  three  pieces  whose  stresses 
are  unknown.  If  a  fourth  piece  is  cut  whose  stress  is  known,  the  moment  of  this  stress  must 
of  course  be  combined  with  the  moments  of  the  other  forces  of  the  system  considered. 

The  preceding  method  is  nothing  more  than  the  method  of  sections  given  in  Art.  33 
except  that  here  the  sum  of  the  moments  of  the  external  forces  is  given  graphically,  and  fur- 
ther, we  use  three  equations  of-moment  for  our  equations  of  equilibrium.  We  may  evidently, 
after  getting  the  stress  in  AD  for  example,  get  the  stresses  in  the  two  other  pieces  by  means 
of  the  other  equations  of  equilibrium,  or  graphically  by  a  force  polygon.  Thus,  7^3  being 
equal  to  the  resultant  of  the  external  forces  R\  /*, ,  and  P^,  if  3-7  =  stress  in  AD,  then  the 
force  polygon  «  3  7  8  gives  7-8  =  stress  in  CD  and  8«  —  stress  in  CE. 

Example  i.  Apply  the  above  method  to  the  bridge-truss  of  Fig.  51.  In  this  case,  since  the  centres 
of  moments  for  the  diagonals  and  verticals  lie  so  far  away.,  it  will  be  better  to  find  the  stresses  in  these 
members  by  putting  2  vert.  comp.  =0,  after  having  found  the  stresses  in  the  chord  members  by  the  use  of 
two -moment  equations. 

49.  Beam  with  Uniform  Load. — Law  of  Variation  of  Moment  (Fig.  58). — Beam  with 

uniform  load  =  p  lbs.  per  foot.  Required  the  moment  of  all  ex- 
ternal forces  upon  either  side  of  any  section,  about  the  neutral 
axis  of  the  section  (called  the  bending  moment  in  the  beam). 

The  equilibrium  polygon  A'cR'  will  be  a  curve  whose  ordi- 
nates,  measured  from  the  closing  line,  will  be  proportional  to 
the  required  moments  ;  and  if  we  make  our  pole  distance,  Om, 
unity,  they  Avill  be  equal  to  them.  To  derive  the  equation  of 
this  curve  we  will  therefore  find  the  law  of  variation  of  these 
moments.  Fig.  58  {a)  shows  the  portion  to  the  left  of  a  section, 
km,  taken  at  a  distance  x  from  the  centre.    The  external  forces 

Their 


are  the  load,  Z^-  —  ^j,  and  the  abutment  reaction,  ^ 


moment  about  a  is 


M  = 


2  \2 


PiL 

2  \2 


P. 


(6) 


For  sign  of  moment,  see  Art.  36. 


32 


MODERN  FRAMED  STRUCTURES. 


d  is  the  panel  length  ;  and  two  loads  at  A  and  B,  each  = 
The  total  load  =  pi  and  the  abutment  reactions  each  =: 


This  is  the  equation  of  a  parabola  with  axis  vertical  and  whose  vertex  is  at  c,  a  distance  =  |^//' 
below  the  origin  c' .  This  parabola  being  drawn,  the  ordinates  from  the  line  A' B'  to  the  curve 
will  give  the  required  moments. 

A  convenient  way  of  drawing  the  parabola  is  to  lay  off  B'b  =  c'c  =  ipP;  then  divide  B'd 
into  any  number  of  equal  parts  and  B'c'  into  the  same  number.  Draw  the  lines  ci,  c2,  etc., 
and  the  verticals  through  i',  2',  etc.  The  intersections  of  corresponding  lines  will  be  points 
on  the  curve. 

50.  Truss  with  Uniform  Load.— Law  of  Variation  of  Moment— In  framed  structures, 

uniform  loads  are  carried  to  joints  usually  by  means  of  sec- 
ondary members,  and  it  is  these  joint  loads  only  which  act 
upon  the  structure  as  a  whole.  For  example,  in  Fig.  59  a 
uniform  load  of  /  lbs.  per  foot  will  be  given  over  to  the  truss 
in  five  concentrations  at  C,  D,  E,  F,  and  G,  each  =  pd,  where 

pd 
2 ' 

Pi 
2 ' 

The  uniform  load  to  the  left  of  any  joint,  as  E,  is  concen- 

pd 

trated  at  the  joints  A,  C,  D,  and  E,  -  being  the  loads  at  A 

and  E  due  to  this  portion  of  the  uniform  load.    The  mo- 
ment of  these  joint  loads  about  E  is  evidently  the  same  as 
the  moment  of  the  uniform  load  to  which  they  are  equiva- 
pl 

lent,  and  the  abutment  reaction  R'  being  equal  to  — ,  we  see  that  the  moment  of  all  the  exter- 
nal forces  to  the  left  of  E  is  given  by  the  ordinate  aa'  to  the  moment  parabola  constructed 
for  the  uniform  load  of  p  lbs.  per  foot.  The  vertices  of  the  equilibrium  polygon  drawji  for  the 
joint  loads,  therefore,  lie  on  this  parabola. 

If  we  take  our  centre  of  moments  at  any  other  point  than  a  point  in  a  load  line,  as  at  m, 
the  mornent  about  m  of  all  the  external  forces  (joint  loads  and  abutment  reaction)  to  the  left 
of  in  is  given  by  the  ordinate  kk' ,  drawn  to  the  proper  segment  of  the  equilibrium  polygon,  and 
nothythe  ordinate to  the  parabola,  as  would  be  the  case  were  the  uniform  load  acting  upon 
a  beam.  We  can  see  this  more  clearly  by  actually  computing  the  moments  in  the  two  cases. 
For  the  truss,  the  moment  can  be  written  in  the  form  x  —  [^pdx -\- pd     {x  —  d)  ^  ^pd  X 

[x  —  2d)]  —  ^pdz.^  For  the  beam,  Fig.  {a),  the  moment  \s  R'  X  x  —  \2pd  X  {x  —  d)]  —  ^ps'', 
The  quantities  within  the  brackets  in  the  two  expressions  are  equal  ;  they  are  the  moment  about 
m  of  the  portion  of  the  load  upon  the  length  AD.  The  moment  upon  the  beam  is  greater, 
therefore,  than  the  moment  upon  the  truss  by  ^pds  —  This  is  seen  to  be  the  bending 

moment  at  m  in  the  secondary  member,  DE,  produced  by  the  uniform  load  in  the  panel. 


Fig.  59. 


ANALYSIS  OF  ROOF-TRUSSES. 


33 


CHAPTER  III. 


ANALYSIS  OF  ROOF-TRUSSES. 


LOADS  AND  REACTIONS. 


51.  Dead  Load. — The  dead  or  fixed  load  supported  by  a  roof-truss  is  made  up  of :  the 
weight  of  the  truss  itself  ;  the  roof,  including  roof-covering,  sheeting,  rafters,  and  purlins  ;  and 
sometimes  the  weight  of  ceilings  and  floors  suspended  from  the  truss.  The  roof  being  designed 
first,  its  weight  can  be  directly  computed,  as  can  be  also  the  weight  of  ceilings  and  floors. 
The  total  weight  of  roof  will  vary  from  5  to  30  lbs.  per  square  foot  of  roof-surface. 

The  weight  of  the  truss  can  only  be  approximated.  From  actual  calculations  it  is  found 
to  be  equal  to  about  i^^l  lbs.  per  square  foot  of  area  covered,  where  /  =  span  length  in  feet ; 
or,  if  ^  =  distance  between  trusses,  then  the  total  weight  of  one  truss  = 


For  short  spans  the  weight  of  the  truss  is  small  compared  with  the  total  load,  and  an  error 
in  its  assumption  is  correspondingly  unimportant.  For  long  spans,  however,  the  error 
becomes  larger,  and  if,  after  a  preliminary  design,  it  is  found  to  be  excessive,  the  weight  must 
be  reassumed  in  accordance  with  the  design  and  the  computations  revised. 

52.  Live  Load. — The  live  or  variable  load  consists  of  the  wind  load,  snow  load,  and  floor 
loads,  if  any.  The  maximum  vvind  pressure  on  a  surface  normal  to  its  direction  is  variously 
estimated  at  from  30  lbs.  to  56  lbs.  per  square  foot.  Some  experiments  made  by  Sir  Benjamin 
Baker  during  the  erection  of  the  Forth  bridge*  indicate  that  the  pressure  per  unit  area  upon 
large  surfaces  is  considerably  less  than  upon  small  surfaces.  The  ratio  of  the  unit  pressure 
upon  an  area  of  square  feet  to  that  upon  an  area  of  300  square  feet  varied  from  1.3  to  2.5. 
being  on  the  average  about  1.5.  The  highest  pressure  recorded  during  the  seven  years  over 
which  the  observations  extended  was  41  lbs.  per  square  foot  upon  the  smaller  surface  and 
27  lbs.  upon  the  larger.  As  the  gales  experienced  in  that  vicinity  are  very  severe,  it  seems 
reasonable  to  a.ssume,  for  ordinary  cases  at  least,  a  wind  pressure  of  45  lbs.  per  square  foot 
upon  small  surfaces  and  30  lbs.  upon  large  ones. 

According  to  experiments  recently  made  upon  Mt.  Washington  by  Asst.  Prof.  C.  F.  Marvin. 
U.  S.  Sig.  Service,!  the  relation  between  wind  pressure  and  velocity  is  given  very  accurately 
by  the  formula  u  =  .004  ;  where  ?/  =  pressure  per  square  foot  and  V  =  velocity  of  the  wind 
in  miles  per  hour.  These  experiments  were  made  upon  surfaces  of  4  and  9  square  feet,  the 
unit  pressures  on  each  being  practically  the  same.  The  difference  in  area  was,  however, 
probably  too  small  to  detect  any  slight  difference  in  unit  pressure  which  may  have  occurred. 
Our  assumed  value  of  45  lbs.  corresponds  by  the  above  formula  to  a  velocity  of  105  miles  per 
hour,  and  thus  would  seem  to  cover  any  case  short  of  a  tornado  which  would  destroy  almost 
any  building  supporting  a  roof. 

The  longitudinal  corriponent  of  the  pressure  of  the  wind  upon  a  roof  is  zero  for  smooth 
roofs  and  nearly  so  for  any.  The  normal  component  is  usuall)-  computed  by  the  empirical 
formula  established  by  Hutton's  experiments,  i.e.. 


(0 


u'  =  u  sin  a  "•84  cos  a  - 


(2) 


*  See  Lon.  Engineering,  Feb.  28,  1 8go. 


\  See  Eng.  A^ews,  Dec.  13,  1890. 


34 


MODERN  FRAMED  STRUCTURES. 


where  ii'  —  normal  component,  u  —  pressure  per  square  foot  on  a  vertical  surface,  and  a  ~ 
angle  of  inclination  of  the  roof  with  the  horizontal.  The  following  values  of  u  ,  for  «  =  30  lbs., 
are  computed  for  various  values  of  a  : 

a             u  a             u'  a  u' 

5°   3-9  25°  16.9  45°  27.1 

10°   7.2  30°  19.9  50°  28.6 

15°  10.5  35°   22.6  55°  29.7 

20°  13.7  40°         .  25. 1  60°  30.0 

For  a  greater  than  60",  u'  is  taken  at  30  lbs. 

The  snow  load  is  estimated,  according  to  the  locality,  at  from  10  lbs.  to  30  lbs.  per  square 
foot  of  horizontal  projection,  the 'weight  of  new  snow  per  cubic  foot  being  from  5  lbs.  to  12 
lbs.  according  to  its  dryness.  Snow  load  need  not  be  considered  where  a  is  greater  than  45°  to 
60°,  depending  on  the  smoothness  of  the  roof. 

The  loads  upon  floors  vary  from  50  lbs.  per  square  foot  to  200  lbs.  or  more,  according  to 
the  use  to  be  made  of  the  building.  In  any  case  the  load  anticipated  should  be  approximated 
as  nearly  as  possible. 

53-  Apex  Loads. — ^The  weight  of  the  roof,  and  the  wind  and  snow  loads,  are  transferred 
to  the  truss  by  means  of  the  purlins.  In  large  roofs  the  purlins  should,  if  possible,  be  placed 
upon  the  trusses  at  the  joints;  but  if  it  is  necessary  to  place  them  between  joints,  the  members 
of  the  upper  chord  supporting  them  must  be  designed  to  resist  as  a  beam  as  well  as  a  com- 
pression member  of  the  truss. 

The  snow  and  roof  loads  being  vertical  and  uniformly  distributed  over  each  panel,  the 
joint  loads  are  each  equal  to  one  half  the  sum  of  the  adjacent  panel  loads.    Thus  the  load  at 
^  b,  Fig.  60,  is  equal  to  one  half  the  panel  load  on  be  plus  one  half  the 

panel  load  on  ab.    The  snow  load  on  the  panel  ab  is  of  course  less 
per  square  foot  of  roof  than  on  be.    The  wind  load  at  b  is  equal  to 
one  half  the  wind  load  on  be  combined  with  one  half  the  wind  load 
on  ab,  the  load  on  each  panel  being  normal  to  the  surface.    If  all 
'       panels  in  one  half  the  truss  lie  in  the  same  plane  and  are  equal,  then 
all  joint  loads  are  equal. 
The  above  applies  only  when  the  purlins  are  placed  at  joints.    If  placed  at  intermediate 
points,  the  loads  on  the  purlins  are  found  as  above  and  divided  between  adjacent  joints  in  the 
inverse  ratio  of  the  distance  of  these  joints  from  the  purlins. 

In  roofs  of  ordinary  span  it  is  usual  to  attach  the  shingles  directly  to  small  angle-iron 
purlins.  In  this  case  the  roof  load  may  be  treated  as  a  uniform  load  upon  the  truss,  both  in 
getting  apex  loads  and  in  computing  the  bending  moment  in  the  upper  chord. 

The  weight  of  the  truss  may  be  considered  as  applied  equally  at  each  of  the  upper  joints. 
54.  Reactions. — For  snow  and  dead  loads  both  reactions  are  vertical.  For  wind  load  the 
reactions  depend  upon  the  manner  of  supporting  the  truss.  If  both  ends  are  fixed,  the  wind 
reactions  are  parallel  to  the  resultant  wind  load  ;  if  one  end  is  free  to  move,  i.e.,  on  rollers  or 
supported  on  a  rocker,  the  reaction  at  this  end  is  vertical  and  that  at  the  fixed  end  follows 
from  the  analysis.  If  one  end  be  fixed  and  the  other  merely  supported  upon  a  smooth  iron 
plate,  the  reaction  at  the  free  end  may  have  a  horizontal  component  equal  to  the  vertical 
component  multiplied  by  the  coefficient  of  friction,  which  is  about  \. 


55.  Forms  of  Trus§e§, 

ing  figures. 


ANALYSIS. 

, — A  few  of  the  standard  forms  of  trusses  are  shown  in  the  adjoin- 


ANALYSIS  OF  ROOF-TRUSSES. 


35 


Fig.  6l  shows  a  French  or,  as  it  is  sometimes  called,  a  Fink  roof-tiuss.     It  is  a  very 

common  and  economical  form  for  trusses  up  to  about  150  feet 
span.    The  struts  be,  de,  etc.,  are  placed  normal  to  the  roof. 

These  with  the  upper  chord  are 
made  either  of  wood  or  iron. 
The  other  members  are  ties  and 
are  made  of  iron. 

Fig.  62. 

Fig.  62  is  a  common  form  for 
wooden  trusses.  The  verticals  are  iron  tie-rods.  A  special  diagram  for  this  foim  of  truss  is 
described  in  Art.  81,  p.  66. 

Fig.  63  is  a  quadrangular  truss  adapted  especially  to  roofs  of  small  rise.  It  is  constructed 
of  iron,  riveted  joints  being  used  for  short  spans  and  pin  connections  for  long  spans. 


Fig.  63. 


Fig.  64. 


The  crescent  or  sickle  truss,  Fig.  64,  with  either  a  single  or  double  system  of  web 
members,  is  a  good  form  for  comparatively  large  spans.  Riveted  iron-work  is  used  throughout. 


 ■ — -N:  \  — ^'-^  '252.'8-!-V 


\     ^    ^    \  \  \  \  \'  1'   /  ,^ 

\  \\  -^  \\\V>\v/;^ 

V  \  \  ^,  \  \  \  \vo  / 


Fig.  65. 

For  very  long  spans,  as  for  train-sheds  and  the  like, 

some  form  of  the  arch-truss  is  often  used.  Fig.  65  is  an 

outline  of  the  arch-truss  of  the  train-shed  of  the  Penn./ 

sylvania  Railroad  .station  at  Jersey  City.  The  arch  is 
hinged  at  A,  B,  and  C.  The  horizontal  thrust  at  the  abutments  is  resisted  by  means  of 
a  tie-rod,  AB,  placed  beneath  the  floor. 


36 


MODERN  FRAMED  STRUCTURES. 


If  an  arch  has  no  hinges,  or  has  but  the  two  hinges  at  the  abutments,  the  stresses  depend 
upon  distortion  as  well  as  upon  the  static  load.    These  forms  are  treated  in  Chapter  XIV. 

56.  Analysis  of  a  French  Truss. — In  the  analysis  of  a  roof-truss  we  must  find  the 
stresses  due  to  dead  load  and  combine  them  with  the  stresses  due  to  live  load  so  as  to  get 
the  greatest  possible  tension  and  compression  in  each  member.  As  there  are  but  three  or 
four  different  possible  loadings  for  roof-trusses,  the  graphical  method  is  well  adapted  to  their 
calculation,  it  being  necessary  to  draw  but  one  diagram  for  each  loading.  Where  the  pieces 
of  a  truss  have  many  different  inclinations  the  analytical  method  is  exceedingly  tedious;  but 
if  that  method  is  preferred,  the  truss  should  be  drawn  to  a  large  scale  and  all  lever-arms 
scaled  from  the  diagram.  The  application  of  the  principles  of  Arts.  32,  33,  will  then  enable 
the  stresses  to  be  readily  computed, 

A  complete  graphical  analysis  of  the  truss  of  Fig.  49,  p.  28,  will  now  be  made.  Span 
=  iooft. ;  rise  =  35  ft;  distance  apart  of  trusses  =  20  ft. ;  roof  divided  into  eight  equal  panels  : 
rollers  at  A.    Length  of  one  side  of  roof  =  1/50'-!-  35"  —  61,0  ft.    Angle  a  —  35°. 


Let  D  =  total  dead  load,  5  =  total  snow  load,  W  —  total  wind  load  upon  either  side, 
^  panel  dead  load,  P,  —  panel  snow  load,  and  P^  —  panel  wind  load. 
From  Art.  5  i,  cq.  (i),  the  weight  of  the  truss  may  be  taken  at  -^-^bP,  =      X  20  X  (100)"  = 
8333  lbs.    Assuming  weight  of  roof  at  15  lbs.  per  square  foot,  the  total  weight  =  15  X  2  X 

61.0  X  20  =  36600  lbs.  Total  dead  load,  or 
D,  =  44933  lbs.,  and  P^  =  5620  lbs. 

Taking  snow  load  at  20  lbs.  per  square 
foot  of  hor.  proj.,  we  have  5  —  40000  lbs. 
and  P,   -  5000  lbs. 

The  normal  component  of  the  wind 
pressure,  according  to  Art.  52,  with  a  —  35°, 
is  22.6  lbs.  per  square  foot.  Whence,  W  = 
22.6  X  20  X  61.0  =  27570  lbs.,  and  P„  — 
\Wz=  6890  lbs. 

After  drawing  the  truss  carefully  to 
scale  (the  scale  used  should  be  from  10  to  20 
feet  to  an  inch),  we  proceed  to  draw  the 
diagram  for  dead  load,  Fig.  67  {a).  Each 
joint  load  ~  P^  —  5620  lbs.,  except  the  loads 
at  the  end  joints,  each  of  which  =  \Pi  —  2810 
lbs.  These  loads  are  laid  off  to  form  the  load 
line  AA' .  The  abutment  reactions  are  A' L 
and  LA,  each  =  k^'A.  Beginning  at  A,  the 
diagram  is  drawn  exactly  as  in  Art.  46,  Fig.  49.  The  scale  actually  used  in  the  calculations 
was  5000  lbs.  to  one  inch.    In  the  diagram,  heavy  lines  denote  compression,  light  lines 


ANALYSrS  OF  ROOF-TKUSSES. 


tension.  The  stresses  in  the  members  due  to  dead  load  are  given  in  the  second  column  of  the 
following  table  ;  they  were  readily  scaled  off  to  the  nearest  hundred  pounds.  The  stress  in 
Z,^  is  given  as  18700  lbs.,  while  by  actual  calculation  it  is  18733  lbs. 

The  diagram  for  snow  load  will  be  a  figure  similar  to  the  one  for  dead  load,  and  the 
stresses  in  the  two  cases  will  be  proportional  to  the  corresponding  loads.  If  we  multiply 
each  dead-load  stress,  therefore,  by  f|-|ff ,  we  will  have  the  corresponding  snow  load  stress. 
These  are  best  found  with  the  slide-rule  ;  they  are  given  in  the  third  column  of  the  table. 

For  wind  stres.ses  we  must  consider  the  wind  blowing  first  from  one  side,  then  from  the 
other,  since  the  abutment  reaction  at  the  roller  end  must  in  both  cases  be  vertical,  and  the 
stresses  produced  in  the  two  cases  will  therefore  not  be  symmetrical.  Fig.  67  {b^  is  the  diagram 
for  wind  from  the  left.  The  load  line  is  AE'.  The  abutment  reaction,  LA,  at  A/\%  easily 
found  by  putting  .2  mom.  about  =0;  the  force  polygon  AE' L  then  gives  the  other 
reaction.  A' L.  Beginning  at  A,  the  diagram  is  readily  constructed.  It  will  be  found  that 
there  are  no  stresses  in  f'e',  e'd\  etc.  Column  4  gives  the  stresses  found  from  the  diagram. 
It  is  seen  that  the  stresses  in  the  pieces  Lg,  gf ,  gd' ,  La',  and  Lc'  are  compressive,  whereas 
they  were  tensile  for  dead  load  ;  the  resultajtt  stresses  are,  however,  all  tensile.  If  the  roof 
had  a  greater  rise,  the  compressive  stresses  due  to  wind  load  would  be  increased  ;  and  if  the 
rise  were  great  enough,  they  would  be  greater  than  those  due  to  dead  load  and  the  resultant 
stresses  would  be  compressive.    These  members  would  then  need  to  be  counter-braced. 

The  diagram  for  wind  pressure  on  the  right  is  given  in  Fig.  67  {c).  The  load  line  is  A' E. 
The  reactions  are  found  as  before  and  the  diagram  then  drawn.  The  stresses  thus  found  are 
given  in  column  5. 

The  maximum  stress  of  each  kind  in  each  piece  is  now  obtained  by  combining  with  the 
dead-load  stress,  whatever  possible  combination  of  the  snow  and  wind  load  stresses  will  give 
the  greatest  total  tension  and  the  greatest  total  compression.  Column  6  gives  these  maximum 
stresses.    It  is  seen  that  no  piece  is  ever  subjected  to  counter-stresses. 


TABLE  OF  STRESSES. 


Dead  Load. 


Snow-load. 


Wind  from  Left. 


Wind  from  Right. 


Maximum. 


+ 
+ 


+  43400 
+  40150 
+  36950 
-f  33700 
-]-  4600 
g20o 
4600 

-  5150 

-  5150 

-  35850 

—  30700 

—  18700 

—  13600 

-  18800 

—  18800 

—  13600 

—  30700 

-  35850 
■  5150 

-  5150 
+  4600 
-j-  9200 
-j-  4600 
+  33700 
+  36950 
+  40150 
+  43400 


+  39100 
+  36150 
+  33250 
+  30300 


-\-  4100 
-h  8300 
4-  4100 

—  4650 

—  4650 

—  32200 

—  27600 

—  16800 

—  12300 

—  i6goo 

—  16900 

—  12300 

—  27600 

—  32250 
4600 
4600 
4100 
8300 

-\-  4100 
-j-  30300 
+  33250 
+  36150 
-(-  39000 


+ 
+ 


+  24500 
-|-  24500 
+  24500 
+  24500 
+  6goo 
4-  13800 


+ 


+ 


6900 
7700 
7700 
18300 
10600 
4400 
14600 
22300 
700 
700 
4800 
4800 


-|-  13600 
-j-  13600 

4"  13600 
-j-  13600 


-f- 18700 
4- 18700 
4"  18700 
-l- 18700 


—  15500 

—  15500 

—  14100 

—  2700 

—  2700 

—  25800 

—  18100 

—  31000 

—  38700 

—  7700 

—  7700 
+  3900 
+  13800 
-j-  6900 
4"  29800 
-j-  29800 
-j-  29800 
4"  29800 


-\- 107000 

4"  looSoo 
+  94700 
-}-  89500 

4"  1 5600 
+  31300 
-(-  15600 

—  17500 

—  17500 

—  86400 

—  73800 

—  49600 

—  40500 

—  58000 

—  61500 

—  44000 

—  89300 

—  106800 

—  17500 

—  17500 
-i-  15600 
+  31300 
4-  15600 
-f  93800 
4- 100000 
4- 106100 
4-  112200 


38 


MODERN  FRAMED  STRUCTURES. 


57-  The  Quadrangular  Truss. ^ — Fig.  68  is  a  half-diagram  of  the  roof-truss  of  the 

Jersey  City  station  of  the  Central  Railroad  of  New 
Jersey.  Distance  between  trusses  =  32.5  ft.  All  upper 
panels  =  13  ft.  in.    Other  dimensions  as  shown. 

Left  end  on  rollers. 

In  this  truss  the  lower  chord  and  diagonals  are  eye- 
bars  capable  of  resisting  tension  only  (except  the  end 
panels  of  the  lower  chord,  which  members  are  counter- 
braced).  Wherever,  therefore,  these  members  would  be 
subjected  to  compression  from  wind  loads,  it  is  necessary 
to  insert  the  counters  (shown  by  dotted  lines)  which  would  then  be  in  tension  and  would 
prevent  the  distortion  of  the  quadrilaterals. 

The  diagram  for  dead  load  is  drawn  as  before,  omitting  the  counters.  Snow  load  stresses 
are  also  found  as  before. 

For  wind  load,  some  of  the  counters  will  come  into  action,  and  as  the  dead  load  always 
acts  with  the  wind  load,  it  will  be  more  convenient  to  combine  the  dead  and  wind  load  for 
each  joint  and  draw  a  diagram  for  this  combined  loading.  Where  there  are  counters,  only 
the  diagonal  which  is  in  tension  should  be  considered.  This  can  easily  be  determined  by  a 
trial  diagram.  If  any  diagonal  be  found  in  compression  where  there  is  no  counter,  a 
counter  must  be  inserted.  The  wind  must  be  considered  as  acting  first  on  one  side,  then  on 
the  other. 

For  maximum  stresses  of  both  kinds  we  have  simply  to  remember  that  the  dead  load 
may  act  alone,  or  combined  with  either  of  the  other  three,  or  combined  with  snow  and  either 
of  the  wind  loads. 

The  student  should  assume  loads  according  to  Arts.  51,  52,  and  make  the  complete 
analysis. 

58.  Analysis  of  a  Crescent  Truss. — Fig.  69  is  a  half  diagram  of  the  roof-truss  of  the 


Fig.  69. 

St.  Louis  Exposition  Building.    The  complete  analysis  is  given  below,  according  to  the  fol 

lowing  data  and  assumptions: 

Distance  between  trusses  =  ^  =  16  ft. 

Total  weight  of  one  truss  —  T  —  ^^bP  —  10400  lbs. 

Weight  of  roof  =  20  lbs.  per  square  foot  of  roof-surface. 

Snow  load  =  20  lbs.  per  square  foot  of  horizontal  projection,  to  be  treated  as  acting: 
first,  all  over  ;  and  second,  on  one  half  only. 


ANALYSIS  OF  ROOF-TRUSSES. 


50 


Wind  pressure  according  to  the  table  of  Art.  52. 

Assuming  an  average  panel  length  of  the  upper  chord  of  9.3  ft.,  the  total  dead  load  per 
panel 

l\  —  20  X  9.3  X  16  +  J^V-^  =  2976  +  743  =  3719-    Call  it  3700  lbs. 

The  snow  load  per  panel  = 

=  20X  8.9  X  16  =  2848.    Call  it  2850  lbs. 

The  wind  load  per  panel  varies  with  the  inclination.  The  different  panel  loads  are  given 
in  Fig.  69,  in  thousands  of  pounds. 

The  complete  dead  load  diagram  is  given  in  Fig.  7o{a)\  the  stresses  for  uniform  snow 


Fig.  70A 


load  are  found  by  multiplying  the  dead  load  stresses  by  ffft-  These  dead  and  snow  load 
stresses  are  marked  D  and  S,  respectively,  in  Fig.  69  and  are  given  in  thousands  of  pounds. 

Fig.  70  (b)  is  the  full  diagram  for  snow  load  on  the  left  side  only.  The  corresponding 
stresses  are  marked  in  Fig.  69.  The  stresses  in  the  members  of  the  left  half  of  the  truss 
due  to  snow  load  on  the  right  only,  are  the  same  as  those  in  the  right  half  due  to  snow 
on  the  left.  These  stresses  are  already  given  in  Fig.  70  {b),  and  are  scaled  off  and  placed 
along  the  pieces  of  Fig.  69  and  marked  Si^.  Only  web  stresses  are  desired  for  unsym- 
metrical  loading,  as  the  chord  stresses  are  greater  with  full  snow-load. 

The  diagrams  for  wind  load  are  given  in  Figs.  70  {c)  and  {d).  Fig.  {c)  is  for  wind  from 
left,  left  end  fixed.  In  drawing  the  diagram  for  wind  from  the  right,  we  may,  for  convenience, 
consider  the  truss  turned  end  for  end.  The  left  end  will  now  be  on  rollers  and  the  right  end 
fixed.  The  vertical  components  of  the  reactions  remain  the  same,  but  the  horizontal 
component  now  acts  at  the  right  end.    The  stresses  found  for  the  right  end  are  thus  really 


40 


MODEKN  FRAMED  HTKUCTURES. 


Fig.  lob. 


Fig,  70f. 


Fig.  -joa. 


ANALYSIS  OF  ROOF- TRUSSES. 


4« 


those  for  the  left  end.  It  is  to  be  noted  that  the  stresses  in  the  right  half  due  to  wind 
pressure  are  not  quite  the  same  as  those  in  the  left  half.  The  stresses  marked  in  Fig.  69  are 
the  maximum  of  each  kind  which  occur  in  either  of  the  two  symmetrical  members  due  to 
wind  from  fixed  end  (Wp)  and  wind  from  roller  end  (H^/?).  Thus,  with  wind  from  either 
direction,  the  greatest  chord  stresses  are  in  the  half  truss  towards  the  wind,  and  these  are  the 
stresses  given.  In  practice  the  truss  would  be  built  symmetrically.  The  total  maximum 
stresses  are  marked  M. 

All  loads  and  stresses  are  given  in  thousands  of  pounds. 

59.  The  Arch-truss. — The  stresses  in  a  three-hinged  arch,  as  in  Fig.  65,  p.  35,  are 
readily  found  by  diagram,  the  reactions  at  A,  B,  and  C  having  been  once  obtained.  Consider- 
ing B  as  the  roller  end,  we  can  get  the  abutment  reactions  at  A  and  B  as  in  any  other  truss, 
either  analytically  or  graphically,  by  treating  the  structure  as  a  whole.  The  stress  in  the  tie 
AB  is  found  by  passing  a  section  through  C  and  the  tie,  treating  the  structure  to  the  left  and 
putting  2  mom.  about  C  —  o.  Knowing  the  stress  in  AB  and  the  abutment  reaction  at 
A,  we  have  at  joint  A  but  two  unknown  forces.  Beginning  then  at  this  point  we  can 
construct  the  diagram  for  the  truss  AC.  Above  the  point  D  a  double  system  of  web 
members  is  inserted  which  is  to  be  treated  as  explained  in  the  next  article. 

The  loads  assumed  in  the  actual  computation  of  the  stresses  in  this  truss  were:*  a  dead 
load  of  about  30  lbs.  per  square  foot  (whether  of  roof  or  of  horizontal  projection  is  not 
stated) ;  a  snow  load  of  17  lbs.  per  square  foot ;  and  a  wind  pressure  of  35  lbs.  per  square  foot 
of  elevation.  The  snow  load  was  assumed  to  exist  :  first,  all  over ;  second,  on  the  twelve 
centre  panels  only  ;  and  third,  on  one  side  only. 

Since  the  two  trusses,  AC  and  CB,  are  supported  alike,  it  is  necessary  to  consider  the 
wind  pressure  from  one  side  only,  the  stresses  in  CB  for  wind  from  the  left  being  the  same  as 
in  AC  for  wind  from  the  right.  For  the  tie  AB,  however,  wind  pressure  on  the  left  increases 
its  tension,  while  wind  from  the  right  decreases  it  and  in  fact  produces  a  slight  compression. 

If  the  tie  AB  is  omitted,  the  horizontal  thrust  must  be  resisted  by  the  abutments  them- 
selves, the  structure  then  being  a  true  arch.  In  that  case  the  abutment  reactions  are  found 
as  in  Art.  31,  Ex.  3.  The  process  there  given  amounts  to  precisely  the  same  thing  as  that 
above  for  finding  the  resultant  reaction  of  tie  and  abutment. 

For  wind  loads,  which  are  so  variously  inclined,  it  will  be  simpler  to  find  the  reactions 
and  also  the  stress  in  tie  AB  by  means  of  an  equilibrium  polygon.  Fig.  65  (a)  shows  a  force 
diagram  for  wind  load  on  the  left.  The  load  line  is  I,  2  ...  14;  the  pole  O  is  chosen  at  any 
convenient  point.  To  draw  the  equilibrium  polygon,  begin  at  A,  the  fixed  end;  the 
closing  line  is  drawn  from  A  to  the  intersection  of  the  last  segment  with  the  vertical  reaction 
Hne  BB'.  The  line  On  drawn  parallel  to  AB',  meeting  the  vertical  14-n  at  n,  determines 
the  reactions  at  B  and  A  ;  they  are  equal  to  14-Ji  and  n-i,  respectively.  Knowing  the 
reaction  at  B,  the  stress  in  AB  is  found  by  a  simple  equation  of  moments,  centre  at  C,  of  the 
forces  acting  on  the  right-hand  portion  ;  or,  the  equilibrium  polygon  may  be  utilized  as  in 
Art.  40,  thus: — Treating  the  forces  {T',  P,,      ,  etc.)  to  the  left  of  a  section  through  C  and 

their  moment  about  may  be  found  by  drawing  Cx'  parallel  to  their  resultant,  «-i4, 
and  x'C  vertically.  Their  moment  about  C  is  then  equal  to  their  moment  about  .1',  which 
equals  x'C  X  sn,  the  horizontal  projection  of  the  ray  O-14.  This  moment  divided  by  the 
lever-arm  oi  AB  gives  the  required  stress.  This  method  is  especially  useful  where  we  com- 
bine the  dead  with  the  wind  loads,  as  in  that  case  the  portion  CB  is  loaded  and  the  analyt- 
ical method  necessitates  the  computation  of  the  moments  of  all  these  loads. 

60.  Trusses  with  Double  Systems  of  Web  Members,  as  in  Figs.  64  and  65,  may  be 
analyzed  by  treating  each  system  separately,  together  with  the  loads  acting  at  the  vertices  of 


*  See  Engineering  News,  Sept.  26,  Oct.  3,  1891. 


42 


MODERN  FRAMED  STRUCTURES. 


that  system.  Thus  in  Fig.  64,  the  chords  with  the  system  shown  by  full  lines  constitute  one 
system,  while  the  chords  with  the  dotted  diagonals  constitute  the  other. 

The  stresses  in  the  diagonals  result  directly,  while  those  in  the  chords  are  found  by 
adding  the  stresses  in  each  member  due  to  each  system.  In  drawing  the  diagrams  it  is 
sufificiently  accurate  to  consider  the  chord  as  a  straight  line  between  consecutive  joints  of  the 
same  system. 

In  Fig.  65  one  system  may  be  taken  as  Da' be' de' fg' hi' kl' C  with  the  radial  struts  at 
a,  c,  e,  g,  i,  and  //  and  the  other  as  DD'ab'cd'e/'gh'ik'lC  with  the  struts  at  b,  d,  f,  h,  and  k. 
In  treating  each  system,  one  half  of  a  panel  load  is  to  be  placed  at  each  apex  and  the  stresses 
determined  as  for  a  simple  truss,  each  piece  being  designed  to  take  either  tension  or 
compression.  One  diagram  must  be  constructed  for  each  system  and  for  each  kind  of 
loading,  and  the  results  combined  for  the  maximum  stresses.  The  maximum  chord  stresses 
result  by  adding  those  due  to  each  system. 


BRIDGE-TRUSSES— ANALYSIS  FOR  UNIFORM  LOADS. 


43 


CHAPTER  IV. 


BRIDGE-TRUSSES— ANALYSIS  FOR  UNIFORM  LOADS. 


GENERAL  CONSIDERATIONS. 


61.  Preliminary  Statement. — The  particular  method  of  analysis  to  be  selected  for  a 
bridge-truss  depends  upon  the  kind  of  loading  and  upon  the  form  of  truss.  The  analysis  of 
bridge-trusses  will  therefore  be  treated  under  three  general  heads  corresponding  to  the  three 
methods  of  loading,  viz.,  Uniform  Loads,  Actual  Wheel  Loads,  and  Conventional  Methods  of 
Loading,  no  distinction  being  made  between  highway  bridges  and  railroad  bridges,  except  as 
to  loads  assumed.  Under  each  of  these  heads  will  be  treated  the  various  forms  of  trusses  for 
which  such  loads  arc  commonly  specified. 

62.  The  Dead  Load  consists  of  the  entire  weight  of  the  bridge,  including  floor  sys- 
tem. For  highway  bridges  the  total  dead  load  per  foot  may  be  taken  from  the  diagram  on  the 
following  page,  prepared  by  plotting  the  weights  per  foot  for  bridges  18  feet  in  width  given  m 
Tables  I,  U,  and  III  in  Waddell's  "  Designing  of  Ordinary  Iron  Highway  Bridges."*  These 
weights  are  from  actual  computations  of  the  bills  of  material  for  sixty  bridges,  and  are  for  the 
ordinary  type  of  single  and  double  intersection  trusses.  "  Class  A"  indicates  bridges  frequently 
subjected  to  heavy  loads;  "Class  B"  indicates  city  bridges  occasionally  subjected  to  heavy 
loads;  and  "Cla.ss  C^"  country  bridges.  The  average  change  in  weight  per  foot  fora  change  m 
width  of  two  feet  is  given  by  the  curves  in  the  lower  part  of  the  diagram  Thus,  for  a  200-ft. 
span.  Class  A,  width  of  roadway  22  ft.,  the  total  weight  per  foot  =  weight  per  foot  for  18-ft. 
roadway  +  2  X  increase  in  weight  for  a  change  in  width  of  two  feet,  =r  960  -I-  2  X  lOO  1  lOO 
lbs.  per  foot. 

For  railroad  bridges  the  dead  load  per  foot  is  given  very  closely  by  the  following  formulae, 
where  /  =  length  of  span  in  feet  and  w  —  dead  load  per  foot  in  pounds,  not  including  the 
track  system  (rails,  ties,  guard-rails,  and  safety-stringers,  if  any). 

For  deck  plate  girders, 


w  =  9/-}-  120. 


(I) 


For  lattice  girders, 


w  =  7/-I-  200. 


(2) 


For  through  pin-connected  iron  bridges  with  steel  eye-bars, 


=  5^+  350- 


(3) 


For  Howe  trusses, 


w  =  6.5/  -|-  275, 


(4) 


*  These  curves  were  first  plotted  by  Prof.  C.  L.  Crandall  in  his  "Notes  on  Bridge  Stresses."  They  are  here 
redrawn  and  sHghtiy  changed. 


44 


MODERN  FRAMED  STRUCTURES. 


These  formulse  are  for  single-track  bridges  ;  for  double-track  add  ninety  per  cent.  The 
weight  per  foot  of  a  single  track  may  be  taken  at  400  lbs.  For  the  load  on  each  truss  take 
one  half  the  above  values. 

Formulae  (i),  (2),  and  (3)  are  of  the  same  form  as  those  in  use  by  a  number  of  the  leading 
bridge  companies,  and  are  based  on  Cooper's  loading  class,  "  Extra  Heavy  A"  (see  page  85). 


/ 

11 

' 

WElCiH  I  S 

/ 

'i 

D 

PER 

UINEAI, 

Foot 

t. 

i 

IRON 

HIGHWAY 

/ 

1, 

K 

f* 

} 

/ 

\ 

k 

r 

- 

A 

i 

Ti 

>C 

U 

7 

- 

r 

— 

1 

,\ 

r 

-H 

1 

] 

T 

n 

— 

)«: 

0 

JJ 

1. 

b 

_ 

: 

r 

p 

Q 

■' 

A 

7 

y 

> 

a 

+ 

I 

\ 

?' 

7 

'J 

> 

1 

H 

'<i 

i 

7J 

1 

- 

f 

71 

> 

H 

^■ 

i< 

t 

r< 

'1 

V 

I 

ii 

i 

•J 

i 

r 

H 

f 

7 

e 

c 

H 

i". 

4 

^ 

e 

/ 

// 

-1 

'< 

c 

t 

r 

t 

r 

f. 

- 

n 

?. 

t 

J 

k 

r 

i 

r 

tj 

-i 

6 

i 

f 

n 

't 

0 

0 

0 

~7t 

Fig.  71. 


Formula  (4)  is  from  assumed  weights  of  Howe  trusses  on  the  Oregon  Pacific  Railroad, 
A.  A.  Schenck,  Chief  Engineer.* 

63.  The  Live  Load  for  highway  bridges  is  usually  taken  as  a  uniform  load  of  from  50 
to  100  lbs.  per  square  foot  of  roadway,  or  the  heaviest  concentrated  load,  due  to  a  road-roller  or 
the  like,  which  is  likely  to  come  upon  the  structure.  The  uniform  load  generally  gives  the 
maximum  stresses  in  the  main  truss  members,  while  for  stringers,  floor-beams,  beam-hangers, 
etc.,  the  concentrated  load  usually  gives  the  greater  stresses.  The  loads  specified  by  Waddell 
are  given  on  the  next  page.    Classes  A,  B,  and  C  have  the  same  signification  as  above. 

For  railroad  bridges  the  load  usually  specified  is  that  due  to  two  of  the  heaviest  locomo- 
tives on  the  road  in  question  when  coupled  in  direct  position,  and  followed  by  a  uniform  load 
due  to  the  heaviest  possible  train.  An  example  of  such  loading  is  given  in  the  next  chapter, 
and  various  standard  loadings  are  given  in  the  chapter  on  Specifications. 


*  See  Engineering  News,  April  26,  1890. 


BRIDGE-TRUSSES— ANALYSIS  FOR  UNIFORM  LOADS. 


45 


op<iii  in  v  cel. 

Moving  Load  per  Square  Foot  of  Floor. 

Classes  A  and  B. 

Class  C 

100  lbs. 

80  lbs. 

50  to  150 

90  " 

80  " 

1 50  to  200 

80  " 

70  " 

200  to  300 

70  ". 

60  " 

300  to  400 

60  " 

50  " 

64.  The  Wind  Pressure  upon  bridges  is  carried  by  horizontal  truss  systems  placed 
between  the  chords  of  the  main  trusses.  The  loads  assumed  and  the  corresponding  stresses 
in  these  trusses  are  discussed  in  Chap.  VII. 

The  stresses  in  the  main  trusses  and  in  these  lateral  systems  due  to  vibration  are  discussed 
in  Part  II. 

65.  Apex  Loads. — The  dead  load  for  short  spans  is  usually  considered  as  applied  at  the 
panel  points  of  the  loaded  chord.  For  long  spans  one  third  may  be  taken  at  the  unloaded 
chord  and  two  thirds  at  the  loaded  chord,  or  the  actual  concentrations  may  be  computed. 

Since  the  live  load  is  given  over  to  the  truss  at  the  panel  points  it  can  thus  afTect  the 
truss  only  at  these  points.  :  The  portion  of  each  end-panel  load  carried  by  the  abutment  does 
not  affect  the  truss  and  need  not  be  taken  into  account  in  finding  either  loads  or  reactions. 
Thus,  for  a  bridge  of  200  ft.  span  and  eight  equal  panels  having  a  uniform  load  of  2000  lbs. 
per  foot,  each  of  the  seven  apex  loads  =  25  X  2000  =  50000  lbs.,  and  each  abutment  reaction 
=  7  X  50000      2  —  175000  lbs. 

66.  Bending  Moment  in  a  Beam.— If  a  beam,  AB,  Fig.  72,  be  loaded  in  any  manner 
and  any  section  taken,  the  sum  of  the  moments  of  the  external 
forces  upon  either  side  of  the  section  about  the  neutral  axis, 
A^,  is  called  the  bending  moment,  or  simplv'  the  mome?it,  at  N. 
From  2  mom.  =  o  we  see  that  the  bending  moment  is  equal 
Dut  of  opposite  sign  to  the  moment  of  the  stress  couple  at  N, 
Fig.  {a).  Whence  we  say  that  the  moment  of  the  external 
forces  is  balanced  by  the  moment  of  the  internal  stresses.  For 
convenience  we  call  bending  moment  positive  when  it  causes 


1 


(a) 


Fig.  72. 

convexity  downwards  or  produces  tension  in  the  lower  fibres,  and  negative  when  the  reverse. 
Its  sign  is  thus  seen  to  agree  with  the  sign  of  the  moment  of  the  external  forces  to  the  left  of 
the  section,  and  to  be  the  opposite  of  the  sign  of  the  moment  of  the  forces  to  the  right. 

The  bending  moment  at  any  point  N  is  always  positive,  the  beam  being  supported  at  the 
ends  ;  for  a  load  to  the  right  of  N  affects  the  forces  on  the  left  only  by  increasing  the  abut- 
ment reaction  and  consequently  the  positive  moment,  while  a  load  to  the  left  of  N  affects  the 
forces  on  the  right  only  by  increasing  the  right  abutment  reaction  and  consequently  the  posi- 
tive bending  moment.  For  a  uniform  load,  therefore,  the  maximum  bending  moment  at 
every  point  occurs  when  the  load  extends  the  whole  length  of  the  beam.  In  Chap.  11,  p.  31, 
we  have  shown  that  the  bending  moment  in  a  beam  under  a  uniform  load  varies  along  the 
beam  as  the  ordinates  to  a  parabola,  the  middle  ordinate  being  =  ^pl'\  where  /  =  load  per  foot 
and  /  =  span.  Also  that  the  moment  at  a  point  any  distance  x  from  the  centre  is  given  by 
the  equjx'ion 

M     kpl' -  ^P^"  (5) 


The  above  equation  may  be  written  in  the  form 


M 


.  (5^) 


46 


MODERN  FRAMED  STRUCTURES. 


That  \s}ythe  bending  moment  at  any  point  in  a  beam  under  a  uniform  load  equals  one  half  the 
load  per  foot  multiplied  by  the  product  of  the  two  segments  into  which  the  beam  ts  divided.  ] 

Equation  (5^)  will  enable  the  moment  to  be  computed  at 
any  point  in  a  beam  or  plate  girder  under  uniform  loading, 
g  For  a  single  concentrated  load,  Fig.  73,  the  maximum 

R2     moment  at  any  point  occurs  when  the  load  is  at  that  point, 
for  a  movement  to  either  side  reduces  the  opposite  abutment 
reaction  and  hence  the  moment.    This  maximum  .moment  is 
Fig.  73.  given  by  the  equation 


„  '  .  ,1 

2  1 

M. 


(6) 


This  is  seen  to  be  the  equation  of  a  parabola  whose  ordinate  is  a  maximum  for  x  =  o,  the 

/ 

value  of  this  ordinate  being  equal  to  P—. 
2P 

If  we  substitute  —  for  p  in  eq.  (5)  we  shall  get  eq.  (6)/thus  showing  that  the  maximum 

bending  moments  due  to  a  single  moving  load  are  the  same  as  for  twice  that  load  when 
uniformly  distributed  over  the  beam.  / 

For  two  equal  loads  a  fixed  distance  apart,  Fig.  74,  the  bending  moment  under  the  left 
hand  load,  P^,  for  example,  is  found  by  adding  the  moments  due  to  each  load.    The  moment 

due  to  P^  is  given  by  eq.  (6).    The  curve  representing  the 
Pi       Pj  variation  in  this  moment  as  the  loads  move  over  the 

beam  is  AcB.    The  moment  at  N  due  to  is 

J-B 
Ir2 


This  can  readily  be  shown  to  be  the  equation  of  a  pa- 
rabola with  maximum  ordinate  =  ~i~  —  —\  when 

I  \2        2  j 

;ir  =  — .    The  curve  Ac' B'  is  this  parabola,  with  vertex 
2 

at  c' .  The  curve  representing  the  sum  of  these  moments  is  evidently  obtained  by  adding  the 
ordinates  of  the  curves  AcB  and  Ac' B\  and  is  ACDB.  The  equation  of  the  portion  ACD  is 
obtained  by  adding  eqs.  (6)  and  (7),  remembering  that  the  loads  are  equal.  This  gives  the 
total  moment 


M: 


P{P         ,     al  ,  \ 

—  I  —  —  2x  \-  ax]  

l\2  2  1 


^8) 


For  a  maximum,  by  putting       —  o,  we  find  x  —        Changing  our  origin  to  this  point 

dx  ■  4 


by  putting  [x'  for  x  in  eq.  (8),  we  have 


„     PIP     al  ,  d 
M  —  -A  2x 

l\2  2^8 


13 


BRIDGE-TRUSSES— ANALYSIS  FOR  UNIFORM  LOADS, 


47 


This  is  the  equation  of  a  parabola,  with  axis  vertical  and  passing  through  the  origin.  The 

P  (  a\' 

maximum  ordinate  is  for  x'  —  o  and  equals  -:^\^  ~  ~  j  >  the  ordinate  OC  in  the  figure.  For 

moment  under  the  right  wheel  the  curve  \s  AD' C B  and  is  symmetrical  to  ACDB.  The  great- 
est moments  for  the  left  half  occur  therefore  under  the  left  wheel,  and  for  the  right  half  under 
the  right  wheel.  The  maximum  moment  in  the  beam  is  at  a  distance  from  the  centre  equal 
to  one-fourth  the  distance  between  the  loads.* 

The  curve  of  maximum  moments  for  any  form  of  loading  may  be  found  by  a  method 
similar  to  the  above,  but  the  subject  will  not  be  treated  further  here.  The  succeeding 
chapter  discusses  the  location  of  the  point  of  maximum  moment  in  a  beam  for  any  number 
of  loads. 

67.  Shear  in  a  Beam. — If  a  section  be  taken  at  any  point  N,  Fig.  75,  in  a  loaded  beam, 
the  sum  of  the  vertical  components  of  the  external  forces  upon  either  side  of  the  section  is 
called  the  shear  on  the  section.  The  sign  of  this  resultant  vertical  force  is  evidently  plus  on 
one  side  and  minus  on  the  other ;  but  for  convenience  we  shall  give  to  the  shear  the  same 
sign  as  that  of  the  resultant  force  on  the  left.  Positive  shear,  then,  is  when  the  left-hand  por- 
tion tends  to  move  upwards  on  the  right,  and  vice  versa.  From  .2  vert.  comp.  =  o  we  know 
that  the  shear  must  be  balanced  by  the  internal  force  or  stress  in  the  section,  that  is,  by  the 
action  of  the  portion  removed  upon  the  portion  considered.  This  stress  is  called  a  sheariiig 
stress  and  in  Fig.  {a)  it  is  replaced  by  the  force  5. 

The  shear  at  any  point  N  in  a  beam.  Fig.  76,  for  a  fixed  uniform  load  of  p  lbs.  per  foot, 

is  equal  to  the  left  abutment  reaction  minus  t'le  load  between  A  and  N,  or 


S=R, -px=p^--x). 


(10) 


This  is  the  equation  of  a  straight  line  having  a  maximum  positive  ordinate  of  /  j  when  x  =  o, 


and  an  equal  negative  ordinate  when  x  —  I.    When  x  =■      S  ■=■  O. 

2 


(a) 


<R2 


Fig.  75. 


— 1 


Fig.  76. 


■■■■ 

— =  iB 

Fig.  77. 


For  a  moving  uniform  load  the  maximum  positive  shear  at  any  point  N,  Fig.  77,  occurs 
when  all  possible  loads  are  added  to  the  right  and  when  there  are  no  loads  on  the  left ;  for 
adding  a  load  to  the  right  increases  the  left  reaction  and  therefore  the  positive  shear,  while 
adding  loads  to  the  left  increases  the  right  reaction  without  affecting  the  other  forces  to  the 
right,  and  hence  decreases  the  positive  shear.    The  maximum  shear  at  TV  is  therefore 


2/  2r 


x)\ 


*  By  equating  the  maximum  bending  moment,  as  above  obtained,  with  the  moment  at  the  centre  when  one  wheel 
is  at  that  point,  it  may  be  shown  that  the  latter  will  be  the  greater  when  a  is  greater  than  0.586/. 


48 


MODERN  FRAMED  STRUCTURES. 


the  equation  of  a  parabola  with  vertex  at  the  right  end.  This  parabola  is  B'C,  BB'  being  the 
axis  ;  the  ordinate  A' C  is  equal  to"^-  The  maximum  negative  shear  is  found  by  loading  to 
the  left  of  the  point,  and  is 


5= = 


(12) 


the  equation  of  the  parabola  A' D. 

Where  a  beam  is  subjected  to  both  a  fixed  and  a  movable  uniform  load,  as  from  dead 
and  live  loads,  the  maximum  positive  and  negative  shears  are  found  by  combining  the  shears 
due  to  each  system  of  loading.  In  Fig.  77,  EF  represents  dead  load  shears  ;  whence  the 
maximum  positive  shears  are  found  graphically  by  adding  the  ordinates  of  this  line  to  those 
of  CB' ,  giving  the  curve  C F.  This  curve  crosses  the  axis  at  G,  the  dead  load  negative  shears 
to  the  right  of  this  point  being  greater  than  the  live  load  positive  shears.  From  G  to  B' , 
therefore,  positive  shear  cannot  occur.  The  curve  ED'  gives  the  maximum  negative  shears 
from  H  to  B' ,  none  being  possible  from  A'  \.o  H.  Between  H  and  G  both  kinds  of  shear  are 
possible.  The  actual  values  of  the  shears  are  best  found  by  the  use  of  eq.  (10)  with  eqs.  (11) 
and  (12).  In  practice  we  need  find  only  the  maximum  positive  shears,  since  the  negative 
shears  are  equal  and  symmetrical  to  them. 

For  a  single  load      ,  Fig.  78,  the  positive  shear  at  any  point  N  is  greatest  when  the  load 

is  just  to  the  right  of  the  point,  for  the  left  reaction  is 
then  a  maximum.    This  maximum  shear  is 


(13) 


the  equation  of  the  straight  line  CB' ,  Fig.  79.  The  ordi- 
nate A' C  —  /*,  .  In  like  manner  the  maximum  negative 
shear  occurs  with  the  load  just  to  the  left  of  the  point 
and  is 


(■4) 


Fk;.  79. 


This  negative  shear  is  represented  by  the  line  A' D. 
For  two  equal  loads,       and  P^,  the  maximum  positive  shear  is  when  P^  is  just  to  the 
right  of  the  point.    The  shear  due  to  P,  is  given  by  eq.  (13)  and  the  line  CB' .    The  shear 
due  to  P^  when  P^  is  at  N,  is  equal  to  left  abutment  reaction  for  /*, ,  or 


S=P. 


I  —  (-r  -f  a) 
1  ' 


05) 


This  is  zero  {ox  x  —  I  —  a,  and  is  equal  to  P^iox  x  —  —  a  if  that  were  possible  ;  it  is  repre- 
sented by  the  line  FE,  where  FB'  and  GA'  —  a  and  GE  =  P., .  The  total  shear  is  the  sum 
of  the  second  members  of  eqs.  (13)  and  (15).  It  is  found  graphically  by  making  CI  —  LA'  and 
drawing  IK  to  meet  CB'  in  the  vertical  through  F.  The  total  negative  shears  are  found 
likewise.    They  are  equal  to  the  ordinates  to  the  line  A'K'I'. 

For  three  loads  the  shear  diagram  would  be  I"K" KB',  and  so  on,  the  curve  approaching 
to  a  parabola  as  the  limiting  case  when  the  load  is  uniform  per  unit  length. 

In  this  article  and  the  preceding,  but  two  of  the  equations  of  equilibrium  have  been  used, 
viz.,  2  mom.  —  o  and  2  vert.  comp.  —  o.  These  two  equations  are  the  only  ones  that 
involve  the  external  forces,  since  these  forces  have  been  taken  as  vertical,  the  usual  condition 
for  bridges  and  beams.  In  the  arch,  however,  the  third  condition  is  involved  and  we  have 
moment,  shear,  and  thrust. 


BRIDGE-TRUSSES—ANALYSIS  FOR   UNIFORM  LOADS. 


49 


BRIDGE-TKUSSKS  WITH  PARALLEL  CFIORDS. 


/J 

p„ 

'r,      j  i 

1 

1  1 

h 

Fig.  8o. 


68.  Chord  Stresses. — If  AB,  Fig.  8o,  be  any  truss  subjected  to  the  vertical  loads 
and  ,  the  stress  in  any  member  3-4  of  the  lower  chord  may  be  found  by  passing  the  section 
Im  cutting  3-4  and  but  two  other  pieces,  treating  the  portion  to  the  left  and  putting  ^  mom. 
about  2  —  0.  The  abutment  reaction  is  supposed  to 
have  been  found  already  by  treating  the  structure  as  a 
whole.  The  moment  of  the  external  forces  to  the  left, 
about  2,  is  called  the  bending  moment  in  the  truss  at  2,  and 
may  be  computed  or  be  scaled  off  from  the  equilibrium 
polygon  drawn  for  the  given  loads.  The  equilibrium  poly- 
gon is  A'abB' ,  and  ce'  X  Om  =  bending  moment.  Its  sign 
is  plus.  The  moment  of  the  stress  in  3-4  must  balance 
this  moment  and  is  therefore  negative,  or  left-handed 
about  2.  The  stress  in  3-4  is  then  tensile  and  equal  to 
the  bending  moment  divided  by  its  lever-arm.  Likewise 
with  the  same  section  and  centre  of  moments  at  3  we  may 
find  the  stress  in  1-2  ;  it  will  be  compression. 

The  bending  moment  at  any  point  between  A  and  B 
is  seen  by  the  equilibrium  polygon  to  be  positive,  as  in  a 
beam.  The  stresses  in  all  members  of  the  lower  chord  are 
therefore  tensile,  and  in  the  upper  chord  compressive. 

Again,  for  maximum  bending  moment  at  any  point  for  a  uniform  load,  F"ig.  81,  the  truss 
must  be  fully  loaded,  but  in  the  case  of  a  truss  the  uniform  load  can  act  only  as  joint  loads. 
We  have  shown  in  Chap.  II,  p.  32,  that  the  vertices  of  the  equilibrium  polygon  for  these  loads 

lie  on  the  parabola  of  moments  drawn  for  a  beam  with  the  .same 
uniform  load.  The  middle  ordinate  =  \pl\  For  the  upper 
or  unloaded  chord,  the  centres  of  moments  are  at  the  loaded 
joints,  and  the  bending  moments  are  therefore  given  by  the 
ordinates  to  the  equilibrium  polygon  or  to  the  parabola.  For 
the  lower  chord  the  centres  of  moments  are  at  the  upper 
joints  and  the  bending  moments  are  giveh  by  ordinates  to  the 
equilibrium  polygon  only.  Thus  for  piece  5-7  the  bending 
moment  at  6  is  given  by  cc' ,  the  ordinate  to  the  parabola  (pole 
distance  =  unity)  ;  but  for  piece  6-i>,  with  centre  of  moments 
at  7,  the  bending  moment  is  given  by  dd' .  Where  the  upper 
joints  arc  over  the  centres  of  the  lower  panels  the  moments  at 
the  upper  joints  are  means  between  the  moments  at  adjacent  lower  joints. 

The  computation  of  the  moments  at  joints  in  the  loaded  chord  is  best  made  by  means  of 
eq.  (5rt),  p.  45,  since  these  moments  are  the  same  as  in  a  beam  with  the  same  loading.  Thus 


the  moment  at  6 


^{A6  X  6j9);  moment  at  4  =  -{AA  X  aB)\  etc.  These  moments  when 
2  2 

divided  by  the  lever-arms  of  the  respective  pieces  give  stresses. 

If  the  panel  lengths  are  all  equal  to  d,  and  m  be  the  number  of  panels  to  the  left  and  ni 

the  number  to  the  right  of  the  centre  of  moments,  then  we  have  from  eq.  (5^)  above  referred  to, 


M  =  %nd  X  ni'd) 


pd'- 


{iiiin')  (16) 


Equation  (16)  applies  equally  well  where  m  and  m'  are  fractional,  provided  the  joint  loads 
are  those  due  to  a  uniform  load. 


5° 


MODERN  FRAMED  STRUCTURES. 


The  moments  at  the  panel  points  of  the  unloaded  chord,  when  these  points  are  not  in 
*:he  same  verticals  as  those  of  the  loaded  chord,  are  best  found  by  proportion,  from  the 
moments  at  the  two  adjacent  joints  of  the  loaded  chord. 

Equation  (i6)  may  be  derived  directly  as  follows: 
Let  AB,  Fig.  82,  be  any  truss  having  equal  panels  in 
the  lower  or  loaded  chord.  Let  ni  be  the  number  of 
panels  to  the  left  of  any  panel  point  C,  and  m'  the 
number  to  the  right.  Let  p  —  load  per  foot.  Panel 
load  =  pd.    There  are  n  —  I  joint  loads  and  therefore 


-nd—l- 
FiG.  82. 


To  the  left  of  C  are  m  —  \  loads 


md 


whose  average  lever-arm  about  C  =  — .    The  bending  moment  at  C  is  therefore 


M  =  R,X  md  —  (m  —  l) pd  X  ^  =  ^^^^         md  —  ^—(m  —  \)m  =  ^—(mm'\  Q.  E.  Do 

2  2  2  2 

69.  Web  Stresses,  Parallel  Chords. — If  AB,  Fig.  83,  be  a  truss  with  parallel  chords, 
the  stress  in  any  web  member,  as  2-3,  may  be  found  by  passing  the  section  cd  and  putting 


Fig.  83. 


^  vert.  comp.  of  the  forces  acting  upon  the  portion  to  the  left  of  the  section,  equal  to  zero. 
The  sum  of  the  vertical  components  of  the  external  forces  is,  as  in  the  case  of  a  beam,  called 
the  shear,  and  must  be  balanced  by  the  vertical  components  of  the  stresses  in  the  members 
cut ;  whence  the  vert.  comp.  of  stress  in  2-3,  Fig.  {a),  is  equal  and  opposite  to  the  shear.  If 
the  shear  is  positive  or  upwards  on  the  left,  then  2-3  is  in  compression  ;  and  if  negative,  then 
it  is  in  tension.  The  vertical  component  in  2-4  is  the  same  as  in  2-3,  since  the  shear  on  the 
section  cutting  2-4  is  the  same  as  that  on  the  section  cd,  no  load  being  at  2.  Likewise  the 
vertical  components  in  7-10  and  7-8  are  equal,  etc.  Since  the  shear  is  constant  between  two 
adjacent  loaded  joints,  we  usually  speak  of  the  shear  in  the  paJtel,  as  shear  in  panel  3-4,  etc. 

For  maximum  positive  live  load  shear  in  panel  3-4,  and  hence  maximum  compression  in 
2-3  and  maximum  tension  in  2-4,  all  joints  to  the  right  of  the  panel  should  be  fully  loaded 
and  all  joints  to  the  left  unloaded.  This  follows  for  the  same  reason  that  was  given  in  the 
case  of  the  beam,  Art.  67.  For  maximum  negative  shear  in  3-4  and  hence  maximum  stresses 
of  the  opposite  kinds  in  2-3  and  2-4,  the  reverse  should  be  the  case. 

For  a  uniform  live  load  the  method  generally  used  in  computing  maximum  shear,  as  in 
panel  3-4  for  example,  is  to  consider  joint  4  fully  loaded  and  joint  3  unloaded.  This  is  evi- 
dently an  impossible  condition,  for  in  order  that  4  may  be  fully  loaded  the  load  must  extend 
to  3,  which  would  then  give  a  half  panel  load  to  3.  The  shears  computed  b)^  this  method  arc 
too  great,  but  't  will  be  the  one  generally  adopted  in  the  following  analysis, 


BRIDGE-TRUSSES— ANALYSIS  FOR   UNIFORM  LOADS. 


A  general  expression  for  the  value  of  this  shear 
may  be  easily  derived. 

Let  m,  Fig.  84,  =  the  number  of  the  panel  in 
question  (3-4)  counting  from  the  left  end,  =  the  a 
number  of  the  last  loaded  joint,  calling  A  zero  ;  and  r 
n  =  the  total  number  of  panels.    The  shear  in  the 
panel  =  the  left  abutment  reaction  =  R^.  Taking 
moments  about  B,  we  have 

X      —  pd[i  +  2  +  3  +  .  .  .  +  {n—in)}d~  o, 
whence  the  maximum  positive  shear  = 


5.  =  i?,  =  —  [l  +  2  =  -  m){n  -  w  +  I). 


(17) 


The  exact  position  for  maximum  shear  will  now  be  derived,  and  the  shears  for  this  posi- 
tion will  be  found  subsequently  for  a  few  cases  and  compared  with  those  found  by  the 
approximate  method. 

Take  the  same  figure  and  notation  as  in  the  above  case.  Let  x  =  distance  from  4  to  the 
head  of  the  load.  As  loads  are  added  to  the  left  of  4,  the  shear  in  the  panel  is  increased  so 
long  as  the  increment  to  the  abutment  reaction  at  A  is  greater  than  the  corresponding  incre- 
ment to  the  panel  reaction  at  3,  since  shear  =  abutment  reaction  minus  load  at  3.  This  is  true 

(n  —  n  —  w  ,  ...        .  , 

—        ^a,  at  which  point  the  increments 


X      (n  —  m)d  4-  X  ., 

until  -3  =  ^  7  ,  or  until  x  — 

a  I  I  —  a  11  —  \ 

of  the  reactions  are  equal.    The  moving  load  should  therefore  extend  to  the  left  of  4  a 


distance  x  — 


 d.    The  shear  for  this  position  in  panel  3-4  is 

<^  —  I 

S  =  R  ^  pl{n  -  m)d  +  xY     px'  ^  pd 

*        '      2<^  2/  2d      2{n  —  i) 


,  (n  —  my. 


(17a) 


This  differs  from  (17)  in  having  the  factor  ~  j-  in  place  of  ,  and  is  thus  seen 

to  give  somewhat  the  smaller  value.    The  actual  difference  is  a  maximum  for  m  =  — or 

at  the  centre,  and  at  that  point  has  a  value  of  ^ .  ^,  or  nearly  same  for  all  spans. 

on  o 


II      2  1 


1  ]  If. 


Fig.  85. 


Notice  that  although  the  loading  by  this  method  is  not  so  simple  as  by  the  approximate 
'Tiethod.  yet  the  expression  for  actual  shear  is  quite  as  easy  of  application  as  formula  (  17). 
70.  The  Warren  Girder. — Fig.  85  shows  a  triangular  truss  or  Warren  girder  as  a  deck 


52 


MODERN  FRAMED  STRUCTURES. 


railroad  bridge.    As  such  it  is  usually  a  riveted  structure  constructed  of  angles  and  plates,  and 
is  used  for  comparatively  short  spans.    The  analysis  of  this  truss  is  as  follows : 

Let  the  span  =  105  ft.  ;  the  height,  h,=  15  ft.,  and  the  panel  length,  =  15  ft.  The 
number  of  panels,  n,  —  7.    The  dead  load  may  be  taken  from  formula  (2),  p.  43.    We  have 

7/  -|-  200  -|-  400 

then,  dead  load  per  foot  per  truss  —  7v  =  ^  =  667  lbs.    It  will  be  considered 

as  uniformly  distributed  along  the  upper  chord  joints  ;  2  and  14  will  thus  receive  three  fourths 
of  a  panel  load  each.  The  live  load  we  will  take  at  1500  lbs.  per  foot  per  truss.  The  dead 
and  live  load  stresses  will  be  found  separately. 

Chord  Stresses  ;  Dead  Load. — For  the  lower-chord  members  the  centres  of  moments  are 
at  the  upper  panel  points,  and  the  stresses  in  these  pieces  result  by  dividing  the  bending 
moments  at  these  points  by  the  height  of  the  truss.    The  moment  at  any  point  is,  by  eq.  (16), 

— -{mm').   The  stress  in  any  lower  chord  member  is  then  equal  to  —^{mm'),  where  m  and  m' 

are  the  number  of  panels  to  the  left  and  right  of  the  centre  of  moments,  calling  0-2  and  14-16 
half  panels. 

Wd''         667  X  ,  1       r   „        .  1  , 

We  have,  — 7-  =  —  =  5002,  whence  the  followmg  chord  stresses : 

2rt  '2i  I  5 

Stress  in  1-3  =  5002     X  ^^-)  =  1250  (i  X  13)  =  16250  lbs. 

Stress  in  3-5  =  1250(3  X  1 1)  =  41250  lbs.     -  T/t  1 S     J(  f*^  <^  ^  ^  d.^Jl^^^ 
Stress  in  5-7  =  1250(5  X  9)   =  56250  lbs.  y^ruier^.  .nr6~:  S'-^'^li^^) 

Stress  in  7-9  1=  1250(7  X  7)   =61250  lbs.      y//j,c/i  ^g</u.Ls  /^trSL3i(^0  , 

t/i'S       /r      J"^  '^^^       ^  ^^^^ 
These  stresses  are  all  taken  out  with  one  setting  of  the  slide-rule.  'sLcai^Ss:  ^  ^jtZers y^sAl)//'Sa^ 

The  stresses  in  9-1 1,  11-13,  and  13-15  are  equal  to  those  111^^^-^*3-5,  1-3,  respec- 
tively.   These  are  all  tensile  stresses. 

The  centres  of  moments  for  the  upper  chord  members  are  at  the  lower  chord  points. 
These  moments  are  means  between  those  at  adjacent  upper  joints,  and  the  lever-arms  all 
being  equal  to  //,  the  actual  stresses  are  like  means  of  the  stresses  in  the  lower  chord  pieces. 
We  have  then  the  following  stresses : 

16250  41250 
Stress  m  2-4  =  — ~  -\-  — —  =  28750  lbs. 

41250  56250 
Stress  m  4-6  —  +  =  48750  lbs. 

.    ^  „      56250  61250 
Stress  m  6-8  =  +  —  5^75° 

etc.  etc. 

These  are  all  compressive  stresses. 

Chord  Stresses  ;  Live  Load. — The  maximum  live  load  chord  stresses  occur  when  the  bridge 
is  fully  loaded.  They  are  found  in  precisely  the  same  way  as  the  dead  load  stresses  above  ;  or 
they  may  be  obtained  by  multipl3nng  the  above  dead  load  stresses  by  the  ratio  of  live  to  dead 
load,  =  this  case.    By  the  latter  method  a  single  setting  of  the  slide-rule  gives  all  the 

stresses.    They  are  as  follows  : 

1-3  =   36580.  2-4  =  64710. 

3-5  =   92870.  4-6  =  109710. 

5-7=126580.  6-8=132210. 
7-9=137840. 


BRIDGE-TRUSSES— ANALYSIS  EOR   UNIFORM  LOADS. 


53 


Web  Stresses. — The  vertical  component  of  the  stress  in  any  web  member  is  equal  to  the 
shear  on  the  section  which  cuts  that  member  and  two  horizontal  chord  pieces.  The  stress  is 
therefore  a  maximum  when  the  shear  is  a  maximum.  The  dead  panel  load  667  X  15  = 
10000  lbs.  Left  reaction,  ,  =  10000  X  (5  +  2  X  f )  2  =  32500  lbs.  This  is  also  the  shear 
in  panel  0-2,  and  being  upwards  on  the  left  it  is  positive.  The  dead  load  shear  in  each  of 
the  other  panels  is  found  by  subtracting  from  the  abutment  reaction  the  loads  on  the  left  of 
the  panel.    We  have  then  the  following  shears  : 


0-2  =  +  32500. 
2-4  =      32500  - 
4-6  —      25000  — 
6-8  =      1 5000  — 


X  10000  =  +  25000. 
10000  =  +  15000. 
10000  =  +  5000. 


The  maximum  positive  live  load  shear  in  any  panel  occurs  when  all  joints  on  the  right  are 
loaded.  For  maximum  shear  in  0-2  the  bridge  is  fully  loaded.  The  panel  live  load  =  1500 
X  15  =  22500  lbs. 

Shear  in  0-2  =  22500  X(5  +  2Xi)-^2  =  73120  lbs. 

For  maximum  shear  in  2-4  all  joints  except  2  are  loaded.  Taking  moments  about  right 
end,  we  have,  as  in  formula  (17): 


Shear  m  2-4  — 


22500 


:(!xi)+i+i+f+i+f] 


143 

3214  X  ^  =  57450. 


Likewise : 


Shear  in  4-6     =  32i4[(t  Xi)  +  f+  f+  f  +  f] 

-  57450  -  3214  X  V-  =  39770. 


Similarly:  Shear  in   6-  8  =  39770  —  3214  X  |  —  25310. 

Shear  in    8-10  =  25310  —  3214  X  |  =  14060. 
Shear  in  10-12  =  14060  —  3214  X  |  —  6030. 
Shear  in  12-14  =  6030   —  3214  X  |  =  1210. 

The  shear  in  12-14  should  be  equal  to  3214(1  X  i)  =  1205,  which  it  is  very  nearly,  thus 
checking  the  work. 

Adding  the  above  shears  to  those  due  to  dead  load,  we  will  have  the  greatest  possible 
positive  shears  for  all  panels.    They  are  as  follows : 


0-2  =  32500+  73120  =  105620. 
2-4  =  25000  +  57450  =  82450. 
4-6  =  1 5000  +  39770  =  54770. 


6-8=       5000  +  25310=  30310. 
8-10  =  —   5000  +  14060  =  9060. 
10-12  =  —  15000  +  6030  =  —  8970. 


We  see  from  this  that  positive  shear  cannot  occur  to  the  right  of  joint  10. 

The  above  shears  multiplied  by  sec  6  give  the  maximum  stresses  in  the  web  members  due 
to  positive  shear.  For  members  inclining  downwards  toward  the  left,  as  1-2,  3-4,  etc.,  the 
stresses  are  the  same  sign  as  the  shear,  that  is,  positive  or  compressive.  Conversely  for  those 
inclining  in  the  other  direction.  Thus  in  Fig.  85  {a),  the  shear  in  panel  4-6  being  upwards  on 
the  left,  the  force  exerted  on  5-6  by  the  right-hand  portion  of  the  truss  must  be  downwards, 
or  5-6  must  be  in  compression.     For  the  same  reason  4-5  is  in  tension. 


54 


MODERN  FRAMED  STRUCTURES. 


Sec  ft' =r  i/jj' _[_  7.5"  15  =  I.I  18.  Multiplying  the  shears  by  this  factor,  we  have  the 
corresponding  stresses : 

1-  2  =  -f-  1 18080.  6-  7  =  —  33890. 

2-  3  =r  -     92180.  7-  8  =  +  33890- 

3-  4  =       92180.  8-  9  =  —  10130. 

4-  5  =  —  61230.  9-10  =  +  10130. 

5-  6  —  -|-  61230. 

The  maximum  negative  shears  and  resulting  stresses  are  equal  and  symmetrical  to  the 
positive  shears  and  stresses.  Just  as  positive  shear  cannot  occur  on  the  right  of  joint  10,  so 
negative  shear  cannot  occur  on  the  left  of  6,  while  between  these  points  both  kinds  are  pos. 
sible.  Members  to  the  right  of  10  and  to  the  left  of  6  are  therefore  subjected  to  but  one  kind 
of  stress,  while  between  6  and  10  they  may  be  subjected  to  either  kind  and  hence  must  be 
counter-braced.    The  stresses  in  the  latter  members  due  to  negative  shear  are: 

6-  7  =  +  10130.  8-  9  =  +  33890. 

7-  8  =  —  10130.  ^10  =  —  33890. 


On  the  left  of  the  centre  the  positive  shear  is  the  greater,  and  on  the  right  the  negative  shear, 
these  being  of  the  same  sign  as  dead  load  shear  ;  the  opposite  kind  in  either  case  is  called 
counter -shear,  and  the  corresponding  stresses  coimter-stresses. 

yi.   The  Howe  Truss  is  shown  in  Fig.  86.    The  chords  and  diagonal  web  members 

are  of  wood,  and  the  verticals  of  iron. 


This  form  of  truss  is  largely  used  where  timber  is 
cheap,  for  both  highway  and  railway  bridges,  and  under 
that  condition  is  very  economical.  Let  us  take  a 
railway  through-bridge  of  144  ft.  span  with  n  =  8  and 


Fig.  86  ^  =  24  ft.  ;     =  1 8  ft.    From  formula  (4),  p.  43,  we  find 

the  dead  load  per  ft.  per  truss,  =  w,  to  be  (6.5  X  144  -f- 
275  -I-400)      2  =  806  lbs.    Assume  the  live  load  at  i  500  lbs.  per  ft.  per  truss  as  before. 

The  dotted  diagonals  are  not  in  action  for  uniform  load,  as  explained  subsequently.  The 
upper  chord  stresses  are  found  by  dividing  the  moments  about  lower  chord  points  by  /i ;  also 
we  have:  stress  in  1-3  =  stress  in  2-4,  stress  in  3-5  =  stress  in  4-6,  etc.,  because  the  moments 
at  3  and  2  are  equal  and  at  4  and  5  ;  or,  from  2  hor.  comp.  =  o  for  the  portion  to  the  left  of  a 
section  cutting  two  chords  and  a  vertical. 
Making  use  of  the  four  simple  formulae, 


live  load  chord  stress 


dead  load  chord  stress  =  —^mm';  [Eq.  (i6)j 

— 7  mm 
2h 

p 

=  dead  load  chord  stress  X  —'■> 

w 

wd 

dead  load  web  stress  =  "('^  +  i  —  2»«)  sec  Q  ;  and 

pd 

live  load  web  stress  =  — {n  —  m){n  —  m -\-  \)  sec  0,      ...    [Eq.  (17)] 


BRIDGE-TRUSSES— ANALYSIS  FOR   UNIFORM  LOADS.  55 

the  computations  can  conveniently  be  put  into  the  following  tabular  form,  which  is  adapted 
to  either  the  Howe,  Pratt,  or  Warren  type  : 

/=  144  ft. ;        18  ft.  ;  ^  =  24  ft. ;  «  =  8  ;  sec  0  for  the  diagonals  =  1.25. 

wd 

«/  =  810  lbs.  per  foot;   — =7290153.; 
p  =  1 500  lbs.  per  foot ; 

—  =  5470  lbs.; 

-=1.85; 
w 

^=1687  lbs. 


TABLE  OF  STRESSES   IN  ONE  TRUSS. 


Chord  Stresses. 

Web  Stresses. 

Memoers. 

Dead  Load. 

Live-load 
Stress  = 

(3)  X  ^  ■ 
w 

Total  = 
(3) +  (4). 

Pa  nel. 

Dead  Load. 

Live  Load. 

Total 

Shear  = 
(8)-H(io). 

Stress  in 
Diagonal  = 
(iij  X  sec  9. 

m  m' . 

Slress  = 

(2)  X  — — 
2/2 

Shear  = 
(7)  X 

2 

{n  —  m)  X 

Shear  = 
(,)x^-^. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

56 
42 
30 
20 
12  . 

10 

I  I 

12 

1-3,  2-4 
3-5,  4-6 
5-7,  6-8 
7-9 

7 
12 

15 
16 

38300 
65  JOO 
82000 
87500 

70900 
121 500 
151900 
162000 

109200 
187IOO 
233900 
249500 

I-  3 
3-  5 
5-  7 
7-  9 
9-1 1 

7 
5 
3 
I 

—  I 

51000 
36400 
21900 
7290 
-  7290 

94500 
70900 
50600 
33750 
20250 

145500 
107300 
72500 
41040 
12960 

18 1900 
134100 
90600 
51300 
16200 

Column  (2)  can  be  written  out  at  once,  and  column  (3)  is  found  by  multiplying  the  numbers 

wd' 

in  (2)  by        ^  which  can  be  done  with  a  single  setting  of  the  slide-rule.     Another  setting 

gives  column  (4),  and  (5)  is  the  sum  of  (3)  and  (4).    Column  (8)  may  be  found  as  indicated,  or 

by  subtracting  wd  four  times  in  succession  from  51000,  the  left  abutment  reaction.  Column 

pd  .  . 

(10)  is  obtained  by  multiplying  (9)  by  the  constant  factor  —  as  given  in  eq.  (17).  The  positive 

shears  have  here  been  found.  Adding  the  dead  load  shears  to  these  we  have  the  maximum 
positive  shears,  column  (11),  in  all  the  panels.  On  the  right  of  the  centre  they  are  the 
counter-shears.  Only  one  panel,  9-1 1,  has  such  shear,  no  positive  shear  being  possible  to  the 
right  of  1 1. 

Multiplying  the  total  shears  by  sec  6,  =  1.25,  we  have  the  stresses  in  the  diagonals.  These 
members  are  designed  to  resist  compression  only,  hence  the  full  diagonals  on  the  left  of  the 
centre  and  the  dotted  one  in  panel  9-1 1  will  be  in  action.  For  negative  shears  the  full  diago- 
nals on  the  right  and  the  dotted  one  in  panel  7-9  are  in  action.  The  diagonals  6-9  and  9-10 
are  counters. 

The  stress  in  any  vertical  is  equal  to  the  shear  on  the  section  cutting  it  and  two  chord 
members,  or  is  equal  to  the  shear  in  the  panel  towards  the  abutment  from  the  vertical. 

The  maximum  stress  in  8-9  is  equal  to  the  maximum  positive  shear  in  panel  7-9,  or  to  a 
full  panel  load,  whichever  is  the  greater.  For  if  this  maximum  positive  shear  is  greater  than 
a  panel  load,  then  the  shear  in  9-1 1  under  the  same  loading  is  also  positive,  and  8-1 1  will  not 
be  in  action,  thus  throwing  all  the  shear  in  7-9  upon  8-9;  and  again,  the  stress  in  8-9  is  always 


56 


MODERN  FRAMED  STRUCTURES. 


equal  to  the  load  at  9  w  heiiever  the  counters  are  not  In  action.  The  maximum  positive  shear 
in  7-9  =  7290  +  33750  =  41040  lbs.,  and  a  full  panel  load  =  14580  +  27000  =  41580  lbs. 
The  latter  value  is  therefore  the  required  stress. 

pd 

The  live  load  shears  computed  by  the  exact  formula,  5=  —  An  —  ni)^,  o.  t;i  are 

2{n  —  i)  /  '  f    3  ' 

as  follows,  beginning  at  the  left:  94500,  69430,  48210,  30860,  17360,  7715.    The  differences 

between  these  and  those  in  the  table  are:  o,  1450,  2420,  2890,  2890,  2410.    The  error  by  the 

approximate  method  is  on  the  safe  side  and  is  not  objectionable,  for,  being  greatest  at  the 

centre,  it  takes  partial  account  of  the  effect  of  impact,  which  is  very  considerable  on  the 

counters.    Being  a  constant  error  for  all  spans  (in  terms  of  panel  load),  it  also  has  a  greater 

relative  influence  in  short  spans,  where  again  impact  has  a  maximum  effect.    The  effect  of 

impact  is  discussed  in  Part  II. 

In  the  above  bridge,  if  one-third  the  dead  load  be  transferred  to  the  upper  panel  points, 
the  chord  stresses  and  diagonal  web  stresses  will  not  be  changed.  The  stresses  in  the  verticals 
will,  however,  each  be  reduced  by  the  amount  of  the  load  transferred. 

In  practice  all  the  dotted  diagonals  are  put  in  for  the  sake  of  rigidity  and  better 
connections.    This  is  not  done  in  metallic  structures. 

72.  The  Pratt  Truss  is  shown  in  Fig.  87  as  a  deck  bridge.  All  parts  are  of  iron  or 
steel,  the  verticals  being  compression  members  and  the  diagonals  tension  members. 


Fig.  87.  Fig.  88. 


Example  i.  Highway  bridge  as  shown  in  Fig.  87;  class  C,  roadway  14  ft.  wide;  /=  105  ft.;  «  =  7; 
^  =  15  ft.    Find  all  the  stresses. 

Fig.  88  shows  the  usual  form  for  through  or  pony  Pratt  trusses.  The  through  or  deck 
Pratt  truss  is  the  standard  form  of  truss  for  both  highway  and  railway  bridges  of  moderate 
spans.  However  it  is  not  generally  used  for  railway  bridges  where  the  span  is  much  less 
than  100  ft. 

The  stresses  in  the  truss  of  Fig.  88  result  readily  by  methods  similar  to  those  already 
illustrated.  The  pieces  2-3  and  lo-ii,  called  hip  verticals,  carry  only  the  loads  at  their  bases 
The  end  pieces  1-2  and  10-13  are  compression  members.  The  maximum  stress  in  6-713 
from  the  stress  in  the  counter  6-9  or  5-6. 

Example  2.  Find  the  stresses  in  all  members  of  the  truss  in  Fig.  88,  a  railway  bridge  of  150  ft.  span; 
=  30  ft. ;  n  =  6;  live  load  =  1700  lbs.  per  foot  per  truss. 


2     4       6      8      10     12     14     16     18     20     33     24  26 


Fig.  89.  Fig.  90. 


Example  3.  Find  the  stresses  in  all  members  of  the  truss  in  Fig.  89,  a  railway  bridge  of  80  ft.  span  ; 
h  =  10'  ft.;  diagonals  inclined  45°  ;  live  load  =  2000  lbs.  per  foot  per  truss.  Take  dead  load  from  formula 
(2),  p.  43- 


BRIDGE-TRUSSES— ANALYSIS  FOR   UNIFORM  LOADS. 


57 


73.  The  Whipple  Truss,  shown  in  Fig.  90,  consists  of  two  simple  Pratt  trusses 
combined.  It  is  in  fact  often  called  a  "  double  intersection"  Pratt  truss.  The  advantage  over 
the  Pratt  for  long  spans  is  in  having  short  panels,  and  yet  an  economical  inclination  for  the 
diagonals  (about  45°).  It  is  used  extensively  for  highway  bridges,  but  for  railway  bridges  it 
has  been  as  a  rule  discarded  in  favor  of  the  Baltimore  truss,  or  for  the  form  shown  in  Fig. 
1 14,  p.  69. 

The  two  systems  of  web  members  distinguished  by  full  and  dotted  lines  are  commonly 
assumed  to  act  independently. 

This  is  not,  however,  strictly  true,  for  the  chords  connecting  the  systems,  while 
allowing  independent  vertical  deflection,  do  not  allow  independent  horizontal  displace- 
ments, so  that  the  loads  on  one  system  affect  to  some  extent  the  form  of,  and  hence  the 
distribution  of  stress  in,  the  other  system.  The  exact  analysis  can  only  be  made  by  the 
theory  of  redundant  members  given  in  Chap.  XV.,  and  even  then  it  depends  upon  the 
adjustment  of  the  counters.  The  assumption  of  independent  systems  is  probably  very  nearly 
correct  and  enables  the  stresses  to  be  statically  determined.  In  any  case  the  question  affects 
the  web  stresses  only,  as  the  chord  stresses  are  a  maximum  for  a  full  load,  and  in  that  case 
the  counters  may,  with  practical  exactness,  be  assumed  as  fully  relieved. 

Let  us  take  a  highway  bridge  with  dimensions  as  in  Fig.  90;  class  A,  roadway  20  ft. 
wide. 

From  the  diagram  p.  44,  dead  load,  iv,  =  lOOO  lbs.  per  foot,  =  500  lbs.  per  foot  per 
truss.    Live  load  for  class  A  =:  80  X  20  =  1600  lbs.  per  foot,  =  800  lbs.  per  foot  per  truss. 

Chord  Stresses. — For  uniform  loads  the  web  members  meeting  the  upper  chord  at  12,  14, 
and  16  are  not  in  action,  there  being  no  shear  in  panel  11-15  of  the  dotted  system.  The 

stress  in  10-18  is  therefore  equal  to  (mom.  at  13)  -i-  h,  =        (6  X  6),  —  67500  lbs.    Stress  in 

8-10  =  stress  in  10-12  minus  hor.  comp.  of  stress  in  10-13  ;  that  in  6-8  —  8-10  minus  hor. 
comp.  in  8-1 1  ;  etc. 

2d 

The  horizontal  component  in  a  diagonal  =  vertical  component  multiplied  by       or,  in 

d 

the  case  of  the  diagonals  2-3  and  23-26,  by  ^.  The  vertical  component  =  shear  in  the  panel 
of  the  system  to  which  the  diagonal  belongs.    We  have  then  for  the  full  system,  since  2d  —h, 

hor.  comp.  lo-i  3  -  \ivd  =  3750; 
"  "  6-  <^  —  ^wd=  1 1250; 
"      "        2-  5  =  ^wd  —  18750. 

For  the  dotted  system, 

hor.  comp.  8-1 1  wd  =  7500; 
"  "  4-  7  2wd  =  1 5000 ; 
"      "      2-  3  =  ^  X  Swd  =  1 1250. 

We  have  then  by  subtraction  the  chord  stresses  as  follows: 


10-12 

=  67500. 

8-10  =  I I-I3 

=  67500  — 

375"  -  - 

63750; 

6-  8  =  9-1 1 

=  63750- 

7500  = 

56250 ; 

4-  6  =  7-9 

=  56250  — 

1 1250  = 

45000 ; 

2-  4  =  5-7 

=  45000 

1 5000  — 

30000. 

The  sum  of  the  hor.  comps.  of  2-5  and  2-3  should  be  equal  to  the  stress  in  2-4,  thus  giving 
a  check  upon  the  work. 


58 


MODERN  FRAMED  STRUCTURES. 


The  live  load  chord  stresses  are  obtained  as  before  by  proportion. 

Web  Stresses.-^T\\&  dead  load  vertical  components,  or  shears  in  the  separate  systems,  are 
given  above. 

For  live  load  shears,       =  2000,  and  we  have  for  the  full  systenn, 

shear  in    i-  5  =  i^pd  x  5)      2  =  30000 ; 
"      "    5-  9  =  2000  (i  +  2  +  3  +  4)  =  20000; 
"      "    9-13  =  2000  (l  +  2  +  3)  =  12000; 
«      «  13-17  =  2000(1 -|- 2)  =  6000. 

pd 

For  dotted  system,  —  =  1000,  and 

shear  in    i-  3  =  i^pd  X  6)     2  =  36000 ; 
"      "    3-  7  =  1000(1  +  3  +  5  +  7  +  9)  =  25000; 

"        "      7-1  I  —  1000(1  -|-  3  -}-  5  4-  7)  16000; 
"        "    II-I5  =:  1000(1  +3  +  5)  =9000; 
"        "    15-19  =  1000(1  +  3)  =  4000. 


Adding  to  the  above  the  dead  load  shears,  we  have  the  stresses  in  the  verticals,  and  the 
vertical  components  of  the  stresses  in  the  diagonals.  Vertical  component,  of  stress  in  the 
counter  14-17  =  shear  in  13-17  —  6000  —  3750  —  2250  lbs.  This  is  also  the  maximum 
compression  in  13-14.  The  vertical  component  of  stress  in  12-15  o*"  1 1-16  =  9000  lbs.  = 
stress  in  11-12  and  15-16.  There  is  no  positive  shear  in  15-19  and  hence  no  counter  is 
needed.    The  stresses  in  1-2  and  25-26  are  equal  to  the  sum  of  the  shears  in  1-3  and  1-5. 

For  an  odd  number  of  panels  the  arrangement  of  diagonals  of  Fig.  92  is  to  be  preferred  to 


Fig.  gi.  Fig.  92. 


that  in  Fig.  91,  as  it  gives  two  web  systems,  each  of  which  is  symmetrical  about  the  centre. 
The  assumption  of  independent  systems  then  gives  a  more  even  and  probably  a  more  nearly 
correct  distribution  of  stress  over  the  two  systems.  In  either  case,  granting  this  assumption, 
the  stresses  are  found  as  in  the  above  example. 

Where  the  arrangement  of  the  end  posts  is  as  in  Fig.  93,  there  is  a  further  ambiguity 


A  B 

Fig.  93. 


from  not  knowing  to  which  system  the  loads  at  A  and  B  belong.  Assuming  them  equally 
divided  between  the  systems,  is  nearly  correct  and  enables  the  stresses  to  be  readily  found. 
The  uncertainty  of  computations  of  stresses  by  the  usual  methods,  in  double  systems, 
constitutes  a  somewhat  serious  defect  for  such  systems,  and  is  one  cause  that  has  led  to  the 
adoption  of  the  forms  referred  to  at  the  beginning  of  this  article. 


BRIDGE-TRUSSES— ANALYSIS  FOR   UNIFORM  LOADS. 


59 


74.  The  Triple  Intersection  Truss  has  been  built  to  some  extent.  It  is  similar  to  the 
Whipple  truss,  but  has  three  instead  of  two  sets  of  web  members.  The  stresses  are  found  in 
the  same  way  as  in  the  Whipple  truss. 

75.  The  Double  Triangular  Truss  shown  in  Fig.  94  has  two  systems  of  triangular 
bracing. 


8         16        8        10       12       H        16       18       20       22       24  28 


\    '\  A        A  / 

\ '  V  v  \/ 

f  \ /  \'  \/  ^/  \ 

I  V  V  V  V 

Fig.  94. 


This  truss  is  used  both  as  a  short-span  riveted  structure  and  as  a  long-span  pin- 
connected  bridge;  the  Memphis  bridge  and  the  Kentucky  and  Indiana  bridge  are  of  this  form, 
though  modified  as  shown  in  Fig.  loi,  p.  61. 

For  chord  stresses  under  a  full  uniform  load  the  pieces  11-14  and  14-15  are  not  in 
action.  The  moment  at  13  divided  by  h  gives  the  stress  in  12-16.  The  upper  chord  stresses 
are  then  found  by  subtracting  succe.ssivel}^  the  sums  of  the  horizontal  components  of  the 
stresses  in  the  two  web  members  which  meet  at  each  panel  point.  These  horizontal 
components  are  found  as  in  Art.  73,  assuming  each  web  system  as  independent.  For  the 
lower  chord  stresses,  that  in  11-13  is  equal  to  12-16  minus  hor.  comp.  of  12-13;  9-1 1  =  11-' 3 
minus  hor.  comp.  of  11-14,  minus  hor.  comp.  lo-ii  ;  etc. 

The  web  stresses  are  readily  found  as  in  a  simple  triangular  truss,  assuming  each  system 
as  independent. 

76.  The  Lattice  Truss,  Fig.  95,  contains  four  web  systems.  It  is  built  only  as  a  short-span 


Fig.  95. 

riveted  structure.  The  web  members  being  riveted  together  at  each  intersection,  the  different 
systems  cannot  act  independently,  and  in  finding  stresses  it  is  usual  to  treat  the  structure  as  a 
beam.  The  maximum  moment  and  shear  is  found  at  several  different  sections;  the  stresses  in 
the  chords  or  flanges  are  found  by  assuming  them  to  take  all  the  moment,  and  those  in  the 
web  members  by  assuming  the  shear  as  equally  divided  among  the  members  cut. 

Example. —  Find  the  chord  and  web  stresses  at  sections  15  ft.  apart  in  a  lattice-truss  of  90  ft.  span ;  live 
load  =  1800  lbs.  per  foot;  dead  load  from  formula  (2),  p.  43;  12  ft.;  panel  length  between  chord 
points  of  the  same  system  =  1 5  ft. 

77.  The  Post  Truss  shown  in  Fig.  96  is  a  special  form  of  the  double  triangular  truss. 


2  4  6  8  10  12  11  16  18 


1  3  5  7  !)  II  13  15  17  19 

Fig.  q6. 


The  lower  chord  is  divided  into  an  odd  number  of  panels,  and  the  upper  chord  contains  one 
less  panel  than  the  lower.  The  posts  are  inclined  by  one-half  of  a  panel  length,  and  the  ties 
by  one  and  one-half  panel  lengths.     The  chord  stresses  are  readily  found  as  before  by 


6o 


MODERN  FRAMED  STRUCTURES. 


Fig.  97 


a!/ 

1  \ 

P2  Pa 

considering  the  web  members  meeting  at  8,  lo,  and  12  as  not  acting  under  a  full  load  if  it  is 
a  through-bridge,  or  the  members  8-ii  and  9-J2  if  a  deck-bridge.  For  web  stresses,  it  is 
impossible  to  separate  the  systems,  as  they  are  connected  at  the  centre.  The  best  that  can  be 
done  is  to  assume  the  systems  to  be  1,  2,  5,  6,  9,  12,  13,  16,  17,  18,  19,  and  i,  2,  3,  4,  7,  8,  11, 
14,  15,  18,  19.  For  the  counters  9-10  and  10-13  the  first  system  becomes  9,  10,  13,  instead 
of  9,  12,  13;  and  for  the  counters  11-12  and  12-15,  if  required,  the  second  system  becomes 
II,  12,  15,  in  the  place  of  11,  14,  15. 

78.  The  Baltimore ,  Truss  and  the  Subdivided  Triangular  Truss. — The  Baltimore 
truss,  Fig.  97,  or  a  modified  form  of  it.  Fig.  114,  p.  69,  is  used  very  generally  for  long  spans 

The  stresses   are    all    easily  deter- 
-r-     mined  ;  the  panel  lengths  are  short, 
while  the  diagonals  have  an  econom- 
ical inclination. 

Chord    Stresses.  —  The  upper 
chord  stresses  are  found  as  usual 
by  taking  centres  of  moments  at  lower-chord  points,  the  dotted 
diagonals  not  being  in  action  for  full  load. 

The  stresses  in  the  lower  chord  members  are  found  by  taking 
moments  about  upper  chord  points.    Thus  for  piece  b'k,  we  pass  a 
section  cutting  2-4,  b'^,  and  b^,  separate  the  portion    to    the  left, 
Fig.   (fl),  and    put        mom.    about  2,  —  o.     We   have,  therefore, 
X  2d  —      X  d  X  d  —      X  h  =  o,  whence  5,  is  determined. 

It  is  to  be  noted  that  the  moment  of  the  external  forces  acting  on 
■   the  portion  considered,  about  the  point  2,  is  not  the  ordinary  "  bending 
moment "  in  the  truss  at  2,  since  here  we  have  included  the  force  P^ 
which  is  on  the  right  of  the  centre  of  moments.    In  the  equilibrium  polygon  this  moment  is 
represented  by  the  ordinate  xy. 

The  stress  in  T^b  is  evidently  equal  to  b^.  The  other  chord  stresses  are  found  similarly  to 
the  above. 

Web  Stresses. — The  stress  in  each  sub-vertical,  aa',  bb',  etc.,  is  equal  to  the  load  at  its 
base  =  {w  -\- p)d.  The  vertical  component  of  the  compressive  stress  in  each  of 
the  pieces,  rt'3,  (^'3,  c'i,,  etc.,  is  found,  by  a  diagram  of  joints  a',  b',  etc.,  to  be  equal 
to  one  half  the  stress  in  the  sub-vertical,  =  ^(w  -\-p)d.  Thus  for  joint  b',  Fig.  98, 
draw  1-2  vertical  and  equal  to  the  stress  in  bb',  then  2-3  parallel  to  3<^',  and  3-4 
and  4-1  parallel  to  2-5,  closing  the  polygon,  the  point  4,  however,  being  unknown. 
The  vertical  component  of  2-3  =  2(1-2)  =  ^{w  -\- p)d. 

The  vertical  components  in  la',  b'i),  c'y,  d'g,  %e' ,  and  lof  are  equal  respec- 
tively to  the  maximum  positive  shears  in  the  panels  to  which  these  pieces  belong,  as  they  are 
the  only  inclined  members  in  the  panels,  which  carry  positive  shear.  The  stresses  in  the 
verticals  4-5,  6-7,  and  8-9  are  equal  to  the  vertical  components  in  4.0',  6d',  and  8/, 
respectively,  plus  whatever  load  may  be  applied  at  the  upper  panel  point.  The  tension  in 
2-3  =  vert.  comp.  in  a'l  -\-  vert.  comp.  in  b'l  -\-  load  at  3,  =  2{w  -)- p)d. 

The  vertical  component  of  the  stress  in  2b'  =  shear  in  ^b  —  vert.  comp. 
in  3//.  To  determine  for  what  nosition  of  the  loads  this  stress  is  a  maxi- 
mum we  note  that  the  addition  of  a  live  panel  load  at  b  increases  the  shear 
in  3(^  by  {fpd,  and  increases  the  vertical  component  in  ^b'  by  ^pd.  The 
vertical  component  in  2b'  is  then  increased  by  (-^  —  ^)pd  —  -f-^pd,  thus  show- 
ing that  the  loads  should  extend  up  to  /;  from  the  right  for  a  maximum 
in  2b' .  With  such  loading,  then,  the  shear  minus  \{w  -\-  p)d  is  the  vertical 
component  of  the  required  stress.    Similarly  for  ^^c'  and  6d'.    The  vertical  component  in 


Fig.  98 


BRIDGE-TRUSSES— ANALYSIS  FOR   UNIFORM  LOADS. 


6i 


a'2  =  shear  in  ^3  -|-  vert.  comp.  a';^,  as  is  seen  from  Fig.  99  by  putting  2  vert.  comp.  =  o. 
Adding /fl^  to  decreases  the  positive  shear  in  ^3  by  3^1^/^,  but  increases  the  vertical  com- 
ponent in  rt'3  by  -f^pd,  thus  increasing  the  vertical  component  in  a'2  by  -x-^pd.  Hence 
for  a  maximum  stress  in  a'2  the  bridge  should  be  fully  loaded. 

When  the  pieces  e'w  and  /'13,  and  similar  members  on  the  left,  act  with  the  counters, 
they  are  in  tension.  The  vertical  component  of  the  tension  in  e'w  =  positive  shear  in 
eii  -\-  vert.  comp.  in  e'lO,  the  piece  ge'  not  being  in  action  and  the  vert,  comp  in  e'lo  being 
equal  to  one  half  the  load  at  e.  A  process  of  reasoning  similar  to  that  above  shows  that  for 
a  maximum  in  e'll  the  load  should  extend  to  II.  The  stress  in /'13  is  found  in  like  manner. 
If  the  shear  in /13  plus  ^zvd  {=  vert.  comp.  in  f'\2)  should  be  negative,  then  there  would  be 
no  tensile  stress  in  /'13. 


\t  / 
i  \ 

1 
1 

Fig.  100. 

Example. — Find  the  stresses  in  all  members  of  the  truss  in  Fig.  100.  Span  =  320  ft.;  h  =  40  ft.;  live 
load  =  1500  lbs.  per  foot ;  dead  load  by  formula  (3),  p.  43.  Consider  one  third  the  dead  load  as  applied  at 
the  upper  chord. 

Fig.  loi  shows  a  method  of  subdividing  the  panels  in  the  double  triangular  truss.  The 
computation  of  stresses  is  but  slightly  altered. 


Fig.  TCI.  Fig.  102. 


The  intermediate  panel  loads  at  a,  b,  c,  etc.,  may  be  considered  as  transferred  to  the  main 
panel  points  1,3,  5,  etc.,  by  means  of  separate  small  trusses  or  trussed  stringers,  irt'3,  3<^'5, 
etc.,  as  shown  in  Fig.  102.  Whatever  stresses  there  may  be  in  the  inclined  and  horizontal 
members  of  these  small  trusses  must  of  course  be  added  to  the  stresses  for  the  same  loading 
in  those  members  of  the  main  truss  with  which  they  in  reality  coincide. 

The  upper-chord  stresses  are  then  found  as  in  Art.  75,  considering  all  loads  as  applied  at 
the  main  panel  points.  The  stresses  in  the  lower  chord  as  part  of  the  main  truss  are  found  in 
the  same  way  ;  to  these  must  be  added  the  stresses  in  the  chord  as  belonging  to  the  trussed 
stringers. 

The  dead-load  web  stresses  are  found  in  a  similar  way  to  the  chord  stresses;  that  is,  by 
treating  the  members  that  are  common  to  the  two  trusses  as  belonging  first  to  one  and  then 
the  other,  and  adding  the  results. 

For  live-load  web  stresses  the  member  6r',  for  example,  receives  its  maximum  tension 
when  the  main  joints  7,  11,  and  15  are  fully  loaded,  which  requires  loads  at  c,  d,  e,  f,  g,  and 
The  load  at  c  affects  the  stress  in  6c'  only  by  adding  to  the  load  at  7,  since  6c'  belongs  only  to 
the  main  truss.  For  the  maximum  tension  in  c'j,  however,  c  should  not  be  loaded,  since  the 
addition  of  this  load  causes  a  compression  in  c'j,  as  part  of  the  small  truss,  which  is  greater 
than  the  additional  tension  produced  in  this  piece  as  part  of  the  main  truss.  Similarly,  for 
the  maximum  compression  in  %c',  joints  d  to  k  are  loaded,  while  for  5^-'  joint  c  is  also  loaded. 

Example. — Find  the  stresses  in  a  deck-hr\6g&  similar  to  the  above  but  with  the  sub-verticals  extending 
upwards  from  the  centre.    Span  =  360  ft. ;  h  =  45  ft. ;  loading  as  in  previous  example. 


62 


MODERN  FRAMED  STRUCTURES. 


BRIDGE  TRUSSES  WITH   INCLINED  CHORDS. 

79.  Chord  Stresses. — Since  the  chord  stresses  are  all  a  maximum  for  full  live  load  they 
are  most  readily  found  graphically,  only  one  diagram  being  necessary. 

The  analytical  method  has  been  sufficiently  indi- 
cated in  Art.  68. 

Instead  of  the  actual  chord  stresses,  it  is  often 
desired  to  get  only  their  horizontal  components. 

In  Fig.  103  the  stress  in  3-5  =  mom.  at  4  4-  \a. 
Draw  the  vertical,  4<a;'.  Then  in  Fig.  {a)  substitute  the 
vertical  and  horizontal  components  for  the  stress  in 
3-5,  applying  them  at  a' .  Then,  since  the  moment  of 
K  about  4  is  zero,  we  have 


H  —  mom.  at  4  -i-  ; 


(18) 


Fig.  103. 

80.  Web  Stresses. 


and  in  general,  the  horizontal  component  of  the  stress 
in  any  chord  member  is  equal  to  the  bending  moment 
at  the  opposite  joint,  divided  by  the  vertical  ordinate 
from  the  joint  to  the  chord. 
With  inclined  chords  the  shear  in  any  panel  is  not  taken  by  the 
web  member  alone,  since  the  chord  stress  has  a  vertical  component. 

Analytically  the  stresses  are  best  found  in  the  verticals,  as  4-5,  Fig.  104,  by  putting 
^  mom.  about  /  =  o,  the  point  /  being  the  in- 
tersection of  the  two  chord  members  cut  by  the 
section  pq  through  the  vertical.  Those  in  the 
inclined  members,  as  2-5,  may  be  found  in  a 
similar  way,  /  being  the  lever-arm  of  2-5  ;  or 
since  t  is  awkward  to  compute,  we  may  find  the 
stress  in  2-5  by  first  finding  the  horizontal  com- 
ponents in  2-4  and  3-5.  Then 


hor.  comp.  2-5  =  hor.  comp.  2-4  —  hor.  comp.  3-5. 


Fig.  104. 


If  hor.  comp.  2-4  >  hor.  comp.  3-5,  then  2-5  must  be  in  tension,  and  vice  versa.  The 
horizontal  components  of  the  chords  are  very  readily  computed  by  eq.  (18),  especially  where 
there  are  verticals. 

For  the  maximum  stress  in  any  web  member  each  joint  up  to  the  section  cutting  the 
member  should  be  fully  loaded.  If  loaded  on  the  longer  segment  of  the  bridge,  this  will  give 
the  main  stress  in  the  member  or  the  stress  in  the  main  diagonal ;  and  if  on  the  shorter  segment, 
it  will  give  the  counter-stress  or  stress  in  the  counter.  This  is  true,  however,  only  when  the 
chord  members  which  are  cut  meet  beyond  the  abutment,  the  usual  condition.  Thus  for  a 
maximum  tension  in  2-5,  Fig.  104,  joints  5,  7,  and  9  should  be  loaded  ;  for  adding  a  load  to 
the  right  of  pq  increases  i?, ,  and  hence  the  negative  moment  about  /  and  stress  in  2-5,  while 
adding  a  load  to  the  left  of  pq,  increases  7?,  less  than  the  load,  hence  increases  the  negative 
moment  about  /  less  than  the  positive  moment  and  therefore  decreases  the  stress  in  2-5. 
Similarly  for  any  other  web  member. 

A  formula  for  the  horizontal  component  of  the  stress  in  a  web  niember  could  easily  be 
written  out,  but  it  would  be  too  complicated  for  ready  use.  Each  case  can  easily  be  worked 
out  for  itself. 

Graphically,  the  dead  load  web  stresses  will  be  found  by  diagram  at  the  same  time  as  tiie 
chord  stresses.    The  counters  are  at  first  to  be  considered  as  not  acting,  and  the  stresses  in 


BRIDGE-TRUSSES— ANALYSIS  FOR   UNIFORM  LOADS. 


63 


the  main  diagonals  found.  Then,  in  order  to  get  the  combined  live  and  dead  load  stresses 
in  the  counters,  we  must  find  the  dead  load  stresses  by  assuming  the  main  diagonals  as 
not  acting.  The  resulting  stresses  in  the  counters  will  be  of  opposite  sign  to  the  live  load 
stresses  and  will  subtract  from  them. 

For  live  load  web  stresses  a  separate  diagram  must  be  drawn  for  each  position  of  the  load, 
a  different  position  being  required  for  each  pair  of  diagonals  meeting  at  the  unloaded  chord. 
This  diagram  need  be  drawn  only  up  to  the  pieces  whose  stresses  are  desired.  The  abbrevi- 
ated diagram  of  Chap.  II,  p.  29,  will  be  found  to  apply  well  here.  The  reactions  for  the 
several  positions  of  the  loads  should  be  first  computed,  then  laid  off  on  the  same  vertical  and 
all  the  diagrams  drawn.  Or  the  live-load  web  stresses  may  be  found  by  a  single  diagram  as 
follows:  Assume  a  left  abutment  reaction  of  some  convenient  amount,  as  loocxx)  lbs.  and. 
with  no  loads  on  the  truss,  begin  at  the  left  and  draw  a  stress  diagram  for  a  little  more  than 
one  half  the  truss,  or  as  far  as  counters  are  required.  The  main  diagonals  are  to  be  consid- 
ered as  acting  on  the  left  of  the  centre,  and  the  counters  on  the  right.  Scale  off  and  tabulate 
the  stresses  thus  found.  Compute  the  actual  reactions  for  the  various  positions  of  the  loads 
required.  Then  to  get  the  maximum  stress  in  any  diagonal,  multiply  the  stress  found  from 
the  diagram  by  the  true  reaction  corresponding  to  the  position  of  loads  for  a  maximum  stress 
in  this  piece,  and  divide  by  looooo.  With  a  slide  rule  this  method  is  very  rapid  and  easy. 
It  is  based  on  the  fact  that  the  diagram  as  drawn  above  and  those  drawn  for  each  position  of 
the  loads  are  similar  figures. 

The  assumption  of  full  joint  loads  on  one  side  of  the  panel  only  is  often  made  as  in 
parallel-chord  trusses.  The  exact  position  of  the  end  of  a  unifoi  m  moving  load  for  maximum 
web  stress  may  be  found  as  follows : 

Let  X,  Fig.  104,  =  the  distance  from  the  panel  point  on  the  right  of  the  section  to  the 
head  of  the  load  ;  m  =  the  number  of  panels  to  the  left  of  this  panel  point,  and  «  =  the  whole 
number  of  panels.  Let  s  =  distance  /i.  Then  the  stress  in  2-5  will  be  increased  by  adding 
loads  to  the  left  of  5  until  we  reach  a  distance  x  from  5,  at  which  point  the  addition  of  a  load 
will  produce  no  additional  stress.    The  stress  in  2-5  due  to  a  load  P  a  distance     from  5  is 


5  = 


P[x  -\-(n-  ni)d'\  1  /  nvi 
 y^s  —  P-^\s  -{-{m  —  j)d] 


Putting  this  equal  to  zero,  we  have 


whence 


X      (n  —  m)d  x 

 j  -s  -  -^s  -\-{m—  \)d\  =  O, 


n  —  m  

«  —  I  -f-  n{tn  —  i)  j- 


Comparing  this  with  the  value  of  x  given  on  p.  5 1,  for  parallel  chords,  we  notice  that  in 

this  last  expression  we  have  the  additional  term  n{m  —  i)-  in  the  denominator.    This  term 

s 

becomes  zero  when  J  =  00  ,  or  for  parallel  chords,  as  should  be  the  case. 


64 


MODERN  FRAMED  STRUCTURES. 


The  corresponding  stress  in  2-5  is  found  by  taking  moments  about  /.  If  is  the  left 
abutment  reaction,  P'  the  panel  load  at  3,  and  /  the  load  per  foot,  we  have  stress  in  2-5  = 


Ky^s-  P'  y^ls^{m-  \)d\ 


Now 


J<i  =  ,   and    P  =  — r. 

2/  2d 

Substituting  the  above  value  of  x  and  reducing,  we  have 

I 


2n        '  ~^ 


n  —  I  -\-  7t{m  —  i) 


(20) 


81.  The  Parabolic  Bowstring  Truss. — In  this  truss,  Fig.  105,  the  lower  chord  is  hori- 
zontal and  the  upper  chord  joints  lie  in  the  arc  of  a  parabola.  The  bracing  may  be  as  in 
Fig.  105,  or  as  shown  in  Fig.  107.  Formerly  this  truss  was  quite  extensively  used,  but  poor 
details  and  the  difificulty  of  making  it  rigid  against  wind-pressure  have  caused  it  to  be  generally 
abandoned.    However,  as  the  analysis  has  several  interesting  features  it  will  be  given. 

Chord  Stresses  (Fig.  105). — The  horizontal  component  in  any  chord  member  is  equal  to 
the  moment  at  the  opposite  joint  divided  by  the  length  of  the  vertical  at  that  joint.    For  full 


load  the  moments  vary  as  the  ordinates  to  a  parabola,  and  likewise  these  verticals  or  lever- 
arms  ;  hence  their  quotient  is  the  same  for  any  chord  member,  and  may  be  written  as  the 
moment  at  the  centre  divided  by  the  centre  height  of  the  truss.  \{  n  —  number  of  panels, 
p  =  load  per  foot,  /i  =  height  at  centre,  and  d  =  panel  length,  we  have 


hor.  comp.  chord  stress 


pd'n' 


(21) 


This  is  of  course  the  actual  stress  throughout  the  lower  chord. 

JVed  Stresses. — For  full  load,  the  horizontal  components  in  the  chords  being  equal,  there 
is  no  stress  in  any  diagonal.    The  corresponding  stress  in  the  verticals  is  {zv  -\-  p)d. 

For  moving  load,  assuming  full  joint  loads  up  to  the  panel  in  question,  the  horizontal 

pd'^n 

component  of  the  maximum  stress  in  any  diagonal  is  constant  and  equal  to  -57-,  as  will  now 

oil 

be  proved. 

Let  m" ,  Fig.  105,  be  the  number  of  panels  from  the  centre  to  the  joint  on  the  right  of 

ft 

the  diagonal  whose  stress  is  required.  For  convenience,  let  n'  =  and  P  =1  pd  =  live  panel 
load. 


BRIDGE-TRUSSES    ANALYSIS  FOR   UNIFORM  LOADS.  65 

Abutment  reaction  —  R  —  F  —r—  =^  F  -,  .  \a) 

'  2n  4«  ^ 

Hon  comp.  2-4  =  R^X  («'  —  m")d  -r-  4-5 

-^■4,  J -p)  

Hor.  comp.  3-5  =      x  («'  —  m"  —  i)d 2-3 

(;^- i)d  _  


Subtracting  {c)  from      and  substituting  the  value  of  i?,  from  {a),  we  have 

hor.comp.  2-5-/'^;--8j-  (22) 

Q.  E.  D. 

The  stress  in  any  vertical,  as  4-5,  is  found  by  taking  moments  about  /.  The  abutment 
reaction  is  equal  to 

R  =  ^{n'^m"){u'  +  m"+  I) 
'  4«'   ^  ' 


By  proportion  we  find 


(n'  -  m"){2-s)  -  in'  -  m"  -  i)(4-5)  , 

(4_5)_(2-3)  ^-  W 

2-3  =       -  ^^^)  =   ^'"^ 0^ 


4- 


Stress  in  4-5  =  — rT'f  tkj  W 

By  substituting  from  (rt'),  {e\  (/),  and  (^)  in  {h\  we  have,  after  reduction, 


Stress  in  4-5  =  P  '—,   =  P\  -7-).  .        «    .  (23) 

^  4«  ^     4«  4«  / 


Now  the  length  of  4-5  =  h  ,  whence 

pdn  n  ~  l  ,  . 

Stress  ni  4-5  =  length  of  4-5  X  ~g^)  minus  the  constant,  pd        .  .    .  (24) 

To  find  the  maximum  stresses  in  all  the  members  of  a  parabolic  bowstring  truss  we  have 

pd'n' 

only  to  draw  the  truss  to  a  scale  such  that  the  length  of  span  =  ,  or  equal  to  the 

horizontal  component  of  the  chord  stress.  The  length  of  each  upper  chord  member 
multiplied  by  «  will  be  the  stress  in  that  member.    The  length  of  each  diagonal  will  be  the 


66 


MODERN  FRAMED  STRUCTURES. 


stress  in  that  diagonal,  for  by  construction  the  horizontal  component  is  equal  to 


pd^n 


And 


finally,  the  length  of  each  vertical  minus  the  constant, /<2'(«  —  \)—-  2n,  is  the  stress  in  that  vertical. 
Counters  are  evidently  required  in  each  panel. 

Another  example  in  which  the  stresses  may  be  measured  directly  from  the  structure 

itself  when  drawn  to  a  proper  scale  is  the  roof-truss 
in  Fig.  io6  when  under  uniform  load.  The  diagonals 
may  easily  be  proved  to  have  a  constant  horizontal 

component  equal  to  ^^i^  which  equals  the  hor.  comp. 

4/^ 


Ri 


of  the  stress  in  6-8-10,  divided  by  —  . 

2 

If  the  span  1-17  is  then  made  equal  to 


the 


Fig.  106. 


>|R2 


truss  drawn  to  scale,  and  the  construction  made  as  in 
the  figure,  the  following  is  true: 

Each  diagonal  is  equal  to  its  stress. 
Each  vertical  is  equal  to  the  stress  in  the  next 
vertical  toward  the  centre  of  the  truss,  assuming  all  loads  to  be  applied  at  the  upper  panel 
points. 

The  hor.  comp.  of  the  stress  in  6-8-10  =  length  of  9-17,  and  the  stress  itself  =  8-17;  the 
stress  in  4-6  =  6'-iy,  that  in  2-4  =  4'-i7,  and  in  1-2  -  2'-i7. 

The  stress  in  7-9  r=  7-17,  that  in  5-7  =  5-17,  that  in  3-5  —  that  in  1-3  =  3-17. 
The  proof  of  the  above  is  left  to  the  student. 

In  the  parabolic  bowstring  with  triangular  bracing,  Fig.  107,  the  ordinates  from  the  lower 


8      «  10 


1  3  5  7  9  11  13  15  17 


Fig.  107. 


joints  to  the  upper  chord  members  are  less  than  the  ordinates  to  the  parabola,  since  each 
chord  member  is  straight  between  joints.  The  horizontal  components  in  the  upper  chord  are 
therefore  not  quite  constant.  For  the  lower  chord,  with  centres  of  moments  at  upper  chord 
points,  the  moments  are  proportional  to  ordinates  from  the  closing  line,  A'B',  to  the  segments 
of  the  equilibrium  polygon  and  not  to  the  parabola;  hence  the  lower  chord  stress  is  not  quite 
constant.  The  actual  horizontal  components  are  readily  computed  by  the  method  already 
explained. 


BRIDGE-TRUSSES— ANALYSIS  FOR    UNIFORM  LOADS.  67 

The  web  stresses  for  moving  load  are  computed  by  taking  the  difference  between  the 
horizontal  components  of  the  stresses  in  the  chord  members. 

Example. — Find  the  stresses  in  the  truss  of  Fig.  107.  Span  =  80  ft.;  ordinate  9a  to  the  parabola  (not 
to  the  chord  member)  =  16  ft.  A  highway  bridge,  class  C.  The  web  members  form  isosceles  triangles 
with  bases  along  the  lower  chord. 

8^.  The  Double  Bowstring  or  Lenticular  Truss,  Fig.  108,  has  both  chords  in  the 


6  ? 


Fig.  108. 


form  of  a  parabola.  The  floor  may  be  supported  along  the  centre  line  1-17,  or  may  be  hung 
below  along  the  line  AB.  In  the  latter  case  the  horizontal  wind-truss  in  the  plane  AB 
prevents  the  swaying  of  the  main  truss  longitudinally.  With  verticals  and  diagonals  as  in  the 
figure,  the  horizontal  component  of  the  chord  stress  is  constant,  for  the  sums  of  the  ordinates 
to  two  parabolas  give  the  ordinates  to  a  third  parabola.  The  stresses  in  the  members  bear 
the  same  relation  to  their  lengths  as  in  the  single  parabolic  truss. 

83.  The  Pegram  Truss,  Fig.  109,  has  several  claims  to  economy  and  general  excellence 
of  design.  Each  chord  consists  of  panels  of  equal  length,  the  upper  chord  panels  being 
shorter  than  the  lower.  The  upper  chord  points  lie  in  the  arc  of  a  circle,  the  chord  of  which, 
2-16,  is  made  about  one  and  one-third  to  one  and  one-half  panel  lengths  shorter  than  the 
span.  The  versed  sine  may  be  so  taken  that  with  an  economical  centre  height  the  lengths  of 
the  posts  will  be  nearly  equal,  or  it  may  be  so  taken  that  they  will  decrease  in  length  toward 
the  ends  where  the  shear  is  great.  In  a  deck-bridge  the  upper  chord  is  made  straight  and  the 
lower  chord  curved.  Having  assumed  the  chord  and  versed  sine  of  the  circular  arc,  the 
coordinates  of  the  joints  are  readily  computed,  each  chord  section  subtending  the  same  angle 
at  the  centre  of  the  circle. 

Let  us  take  as  an  example  a  200-ft.  through-span,  Fig.  109,  with  seven  panels,  each  equal 
to  28.57  The  coordinates  of  the  upper  paj^el  points  are  given  in  the  figure.  These  points 
lie  in  a  circular  arc  with  a  chord  of  160  ft.  and  versed  sine  of  15  ft.  Each  top  chord  member 
between  pins  is  23.55  ft.  long  except  the  centre  one,  which  is  -^-^  less  or  22.37  f*.  long.  This 
is  made  shorter  to  enable  the  chord  sections  between  splices  to  be  of  uniform  length,  the 
splices  being  towards  the  end  of  the  truss  from  the  pin-points. 

The  dead  load  by  formula  (3),  p.  43,  will  be  equal  to  ^^~^35<3  +  400  __        j^^^  ^^^^ 

per  truss.  The  live  load  we  will  take  at  r8oo  lbs.  per  foot  per  truss.  The  dead  panel 
load  =  875X28.57  =  25000  lbs.  Live  panel  load  =  1800  X  28.57  =  5I430  lbs.  The  stresses 
will  be  found  by  diagram. 

Dead  Load  Stresses. — The  abutment  reaction,  7?, ,  =  3  x  25000  =  75000  lbs.  Laying  off 
BA,  Fig.  Ill,  equal  to  this,  and  AP,  PQ,  and  QR  each  equal  to  25000  lbs.,  we  draw  the 
diagram  for  one  half  the  truss  as  in  Chap.  II.    The  dotted  diagonals  are  considered  as  not  in 


68 


MODERN  FRAMED  STRUCTURES. 


action  for  uniform  load,  but  in  order  to  get  the  stress  in  the  counter  5-8  or  lo-ll  due  to  dead 

and  live  load  we  must  here  draw  the  diagram  first  with  6-7  in  action  and  then  with  5-8  in 

action.    The  resulting  compressive  stress  in  5-8,  G' H '  in  the  diagram,  is  afterwards  combined 

with  the  maximum  tension  due  to  live  load.    The  resulting  stresses  as  scaled  off  from  the 

diagram  are  written  along  each  member  in  Fig.  no,  and  are  marked  "Z>."    For  a  check  the 

•    o       •    ,  ,      875  X  (28.57)'  X  12 

stress  m  8-10  is,  by  moments,  equal  to  — r  =  1 10700  lbs. 

2  X  3°'72 


Fig.  1 10.  Fig.  hi. 


Live  Load  Stresses. — The  stresses  in  the  chords  and  in  1-2  and  2-3  are  a  maximum  for 
full  load,  and  may  therefore  be  obtained  by  multiplying  the  corresponding  dead  load  stresses 

hv  18  0  0 

For  a  maximum  in  3-4  and  4-5,  all  joints  up  to  5  should  be  loaded.    The  reaction  is 
pd 

then  equal  to  ^(i  +  2  +  3+4+5)  =  1 10200  lbs.     Laying  this  off  as  BA^,  Fig,  112,  we 

■proceed  to  draw  the  diagram  as  far  as  piece  4-5  by  the  method  explained  in  Art.  47,  p.  29. 
Substituting  the  triangle  1-4-5  fo''  the  original  framework  we  find  the  stress  in  4-5,  or  EF,  by 
drawing  A^E  parallel  to  1-5  and  BE  parallel  to  4-1  ;  then  EF  parallel  to  4-5  and  BF  parallel 
to  4-6;  whence  EF\s  the  required  stress  in  4-5.  To  find  the  stress  in  3-4,  draw  the  diagram 
for  joint  4  of  the  original  truss.  This  diagram  is  BFEDB,  the  portion  BEE  being  already 
drawn  ;  ED  is  the  stress  in  3-4. 

For  a  maximum  in  5-6  and  6-7  all  joints  but  3  and  5  should  be  loaded.    The  reaction 
pd 

R,  =  — (i  +  2  +  3  +  4)  =  73470  lbs.    Laying  off  BA,  equal  to  this,  we  proceed  as  for  3-4  and 


BRIDGK-TRUSSES— ANALYSIS  FOR   UNIFORM  LOADS. 


69 


4-5.  Substituting  tlie  triangle  1-6-7  fo''  tlie  portion  of  the  truss  to  the  left  of  7,  the  diagram 
BAfi  and  thence  GBH  determines  the  stress  in  6-7,  or  GH.  The  diagram  for  joint  6  is  all 
drawn  except  the  line  OF' .  This  drawn  gives  the  stress  in  the  post  5-6.  The  stresses  in  the 
other  web  members  are  found  in  like  manner.  The  last  loaded  panel  in  each  case  is  indicated 
by  the  subscript  to  the  letter  A  in  the  diagram.  The  stresses  are  given  in  Fig.  1 10, 
marked  "  Z." 

The  live  load  web  stresses  may  be  otherwise  found  by  diagram  as  explained  in  Art.  80, 
p.  62.  That  is,  by  assuming  a  reaction  of  100000  lbs.,  drawing  the  corresponding  diagram, 
and  finding  the  actual  stresses  from  this  diagram  by  proportion.  Fig.  113  is  such  a  diagram, 
with  BA  —  100000  lbs.  by  scale.    The  few  computations  may  be  tabulated  thus: 


LIVE  LOAD  WEB  STRESSES. 


Member. 

Stress  from 
Diagram. 
=  100000  lbs. 

Actual 
Reaction. 

Actual 
Stress. 

I 

2 

3 

4 

3-4 

72,100 

110,200 

79.500 

4-5 

72,500 

1 10,200 

79.900 

5-6 

67,000 

73.500 

49,200 

6-7 

88,700 

73.500 

65,200 

7-8 

77.500 

44, 100 

34,200 

8-9 

121,000 

44,100 

53.200 

lO-I  I 

185,000 

22,000 

40, 700 

Column  (3)  contains  the  actual  reactions  when  the  truss  is  loaded  so  as  to  produce  the 
maximum  stresses  in  the  corresponding  members  of  column  (i).  Column  (4)  is  obtained  by 
multiplying  the  quantities  in  column  (2)  by  those  in  (3)  and  dividing  by  lOOOOO.  The  result- 
ing stresses  should  be  the  same  as  those  found  from  Fig.  112. 

If  analytical  methods  are  preferred,  the  same  general  methods  are  to  be  used  as  given  in 
Art.  79,  i.e.,  the  chord  stresses  found  by  moments  and  the  web  stresses  by  subtracting  hori- 
zontal components  of  chord  stresses.  In  panel  5-7  the  compression  in  5-8  is  to  be  found  for 
dead  load  by  assuming  6-7  as  not  acting,  for  the  same  reason  as  given  in  the  above  analysis.* 

84.  The  Petit  Truss  shown  in  Fig.  114  is  the  standard  form  for  very  long  spans.    It  is 


Fig.  114. 


very  similar  to  the  Baltimore  truss,  the  only  difference  being  in  the  inclined  upper  chord, 
which  is  a  more  economical  arrangement  for  long  spans.  The  pieces  shown  by  dotted  lines 
serve  merely  to  support  the  chords  and  posts  at  intermediate  points,  and  form  no  part  of  the 


*  For  a  full  description  of  the  Pegram  truss,  see  Engineering  News,  Dec.  10  and  17,  1887.  For  illustrations  of 
details  of  three  such  trusses,  including  the  one  above  analyzed,  see  Engineering  A'e2vs,  Feb.  14,  1891. 


70 


MODERN  FRAMED  STRUCTURES. 


truss  proper.  The  vertical  ones  may  be  designed  to  carry  the  weight  of  the  upper  chord  ; 
the  horizontal  ones  have  no  definite  load  and  are  made  of  uniform  size,  sufficiently  strong  to 
resist  in  either  direction  the  buckling  of  the  posts.  Omitting  these  members,  the  analysis 
offers  no  special  difificulties,  as  the  variation  from  the  Baltimore  truss  due  to  inclined  chords 
is  easily  taken  into  account.  If  a  diagram  is  used,  the  fact  that  the  vertical  components  of 
the  stresses  in  ^'3,  f'5,  etc.,  are  each  equal  to  one  half  of  a  panel  load,  enables  the 
diagrams  for  joints  3,  5,  7,  etc.,  as  these  points  are  reached,  to  be  readily  constructed. 

Example. — Find  the  stresses  in  the  truss  of  Fig.  114,  a  double-track  railroad-bridge  with  assumed  live 
load  of  3000  lbs.  per  foot  per  truss. 

85.  Double  Systems. — In  double-intersection  trusses,  with  curved  upper  chords,  each 
system  is  affected  by  loads  on  the  other  owing  to  the  rising  tendency  of  each  angle  of  the 
upper  chord  whenever  there  is  any  stress  in  the  chord. 

Fig.  1 1  5  shows  a  double-intersection  Pegram  truss,  which  will  serve  as  a  general  example 
of  the  forms  under  discussion.  The  span  is  336  ft. ;  length  of  panel  24  ft.  ;  height  at  centre 
45  ft.,  and  at  ends  32  ft.  The  pieces  2-3  and  27-30  are  vertical,  thus  making  the  distance 
2-30  (the  chord  of  the  circular  arc)  equal  to  288  ft. ;  the  versed  sine  =  45  —  32  =  13  ft.  Ail 
upper  chord  panels  are  euqal.  For  dead  load  or  full  live  load,  the  diagonals  meeting  the 
upper  chord  at  14,  16,  and  18  are  assumed  as  not  acting.    The  stresses  for  such  loading  are 


8 


I      A     3      Q     5     K     7       N      9     Q     n     T     13      V     15      V'  17     T'     19     Q'  21  23  25  27  29 


Fig.  115. 

then  readily  found,  either  analytically  or  graphically,  by  commencing  at  the  centre,  finding 
12-14-16-18-20  by  moments,  and  then  passing  towards  the  end.    The  graphical  method  is 


5  X  336  +  350  4"  400 

much  the  better  here.    The  diagram  for  dead  load,  taken  at  ^  =1215 

lbs.  per  foot,  is  given  in  Fig.  117.  After  having  found  12-20  by  moments  the  diagrams  for 
joints  16  and  14  were  drawn,  thus  getting  the  tensions  in  15-16  and  13-14.    Then  assuming- 


BRIDGE-TRUSSES-  ANALYSIS  FOR  UNIFORM  LOADS. 


7> 


12-15  and  1  5  20  equally  stressed  the  diagram  for  15  was  drawn;  then  for  joints  13,  12,  11,  10, 

9,  etc.  The  diagram  was  also  drawn  for  the 
counters,  1 1-16  and  9-14,  acting,  as  was  done 
in  the  previous  examples.  This  part  of  the 
diagram  is  shown  to  a  larger  scale  in  the 
lower  left-hand  corner  of  Fig.  117.  The 
resulting  stresses,  marked  "Z),"  are  written 
along  the  corresponding  members  in  Fig.  1 16 
The  live  load  chord  stresses  were  found  by 
proportion,  making  a  single  setting  of  the 
slide-rule,  assuming  a  live  load  of  i  ^QO  lbs. 
per  foot  per  truss. 

p' 


Fig.  ii8«. 


Fig.  118*. 


For  maximum  live  load  vi^eb  stresses  it  is  sufficiently  accurate  to  treat  the  systems  as 
independent  and  assume  the  chord  members  straight  between  joints  of  the  same  system. 
I'he  maximum  stresses  are  then  found  as  in  a  single-intersection  truss,  a  set  of  diagrams 
being  constructed  for  each  system. 

Fig.  ii8rt  is  the  complete  diagram  for  the  full  system  and  Fig.  Il8^  that  for  the  dotted 
system.    The  stresses  are  marked  "  L  "  in  Fig.  1 16. 


SKEW-BRIDGES. 


86.  Skew-bridges  are  those  in  which  one  or  both  end-supports  of  one  truss  are  not 
directly  opposite  to  those  of  the  other.     Fig.  120  is  a  plan  and  Figs.  119  and  121  are 

elevations  of  the  two  trusses  of  such  a  bridge.  The 
intermediate  panel  points  are  usually  placed  opposite, 
in  the  two  trusses,  so  that  all  floor-beams  are  at 
right  angles  to  the  line  of  the  truss.  Where  the 
skew  is  not  exactly  one  panel,  as  at  the  left  end,  it 
is  necessary  to  move  the  point  K  backward  and  K ' 
forward  in  order  that  AKK'A'  may  be  a  plane  figure^ 


Fig.  119. 


K'B* 


c 


A  BIS 


E' 


6 

c 

d 

e 

f 

\ 

A  

D  E 

Fig.  120. 


The  hip  verticals  are  thus  slightly  inclined,  and  loads  at  their  bases  will  affect  the  lower  chords 
directly. 


72 


MODERN  FRAMED  STRUCTURES. 


In  the  analysis,  each  truss  must  be  treated  separately  unless  the  skew  is  the  same  at  each 
end  and  the  trusses  therefore  symmetrical.  As  far  as  the  load  on  the  trusses  is  concerned,  it 
may  be  assumed  as  applied  along  the  centre  line  XY.    A  full  floor-beam  load  is  equal  to  one 

fh 

panel  load  except  for  GH'  and  BB' .    In  the  former  case  it  is  pd  X  ^  and  in  the  latter 


ac 


All  floor-beam  loads  are  divided  equally  between  the  trusses.    For  any  particular 


loading  it  will  be  necessary  to  compute  actual  joint  loads  according  to  the  above  principles, 
since  in  no  case  are  they  the  same  as  would  be  the  case  in  a  square  bridge.  With  the  joint 
loads  computed  the  analysis  is  simple. 

86a.  The  Ferris  Wheel. — Problem:  To  find  the  stresses  in  the  rim  and  spokes  of  a  wheel  supported 
at  the  centre  and  loaded  with  equal  loads,  W,  placed  at  each  of  the  joints  of  the  rim. 

Let  there  be  36  segments  as  in  the  well-known  Ferris 
wheel ;  let  r  —  radius  and  a.  —  angle  between  consecutive 
spokes,  —  10°.    We  will  consider  two  cases: 

1st.  When  the  spokes  are  rods  capable  of  resisting  tension 
only. — In  this  case  it  will  be  assumed  that  the  spokes  have 
such  initial  tension  in  them  that  when  the  loads  are  applied 
all  will  still  be  in  tension  except  the  spoke  a,  whose  stress  will 
be  reduced  just  to  zero. 

Let  S\ ,  S-i,  etc.,  be  the  stresses  in  segments  i,  2,  etc.,  of 
the  rim,  and  Sa,  Si,  etc.,  be  the  stresses  in  spokes  a,  b,  etc. 
Treating  the  joint  at  the  top  of  the  wheel  as  free  and  remember- 

a 

ing  that  Sa  —  o,  we  have  {S-,  -f  ^ae)  sm  IV  =  o,  or  smce 


The  stress  in  any  other  segment, 


Fig.  i2ia. — The  Ferris  Wheel. 


Jbi  —  636 ,  Oi  =  —  cosec  -. 

2  2 

as  Sn,  may  be  found  by  passing  the  section  pg',  treating  the 
portion  to  the  right  and  taking  moments  about  the  centre  of 
the  wheel.    This  gives 


Ser  —  S,r  +  Wr  sin  a  -j-  Wr  sin  2a  -|-  .  .  .  -|-  Wr  sin  7a,* 
hence  Sa  =  w{\  cosec     -t-  sin  a  -f-  sin  2n:  -|-  .  .  .  -l-  sin  7aj. 

The  segment  having  the  greatest  stress  is  No.  18,  and  the  value  of  this  stress  is 
5, 


The  tension  in  any  spoke,  as  h,  is  Sh 


i  cosec  —  -I-  sin  ttr  -1-  sin  2a  -I-  .  .  .  4-  sin  17a)  =  \  j.i6lV. 


(Si  +  Ss)  sin  W^cos  /J,  where  ;.  is  the  inclination  of  spoke  /i  to 


the  vertical.    The  spoke  having  the  maximum  tension  is  /  and  its  stress  is  St  =  {Sis  +  S19)  sin  —  +  IV  =  4. 

The  initial  tension  required  to  produce  the  conditions  assumed  above  is  2  JV  in  each  spoke,  as  may  be 
proved  thus:  By  symmetry,  if  we  apply  loads  of  W  upwards,  the  tension  in  a  will  be  4 and  the  stress  in 
/  will  be  zero;  hence  if  we  apply  loads  of  /Fboth  upwards  and  downwards  (equivalent  to  removing  all  loads) 
the  stresses  in  a  and  t  will  be  means  between  these  caused  by  the  extremes  of  load,  or  the  stress  in  both  a 
and  /  will  be  2  W,  and  hence  2  W  in  all  other  spokes. 

2d.  When  the  spokes  are  stiff  members  and  put  in  luitliotit  initial  stress. — The  tension  in  any  spoke  will 

in  this  case  evidently  be  the  same  as  in  the  first  case,  minus  the  initial  tension  o{  iW;  that  is,  the  stress  in 

a  will  he  2  IV  compression  and  that  in  /  will  be  2  IV  tension.    The  stress  in  aiiv  segment  of  the  rim  is  the 

same  as  in  the  first  case,  minus  the  stress  caused  by  the  initial  tension  of  2  IF  in  the  spokes,  or  minus 

a  IV  a 

fFcosec  — .    The  stress  in  segment  i  will  then  be  —  cosec  —  =  11.48  J'F  tension,  and  the  stress  in  segment 


18  will  be  .b'lB  =  IVi  —  i  cosec  — I-  sin  a  +  sin  2a  -f  .  .  .  +  sin  17a   =  (17.16  —  1 1.48)  IV  =  5.68  compres- 


sion. The  above  are  the  maximum  and  minimum  stresses  occurring  in  the  spokes  and  rim  for  this  case. 
The  wind  stresses  are  readily  obtained,  since  each  joint  is  supported  separately  by  diagonals  from  the  axle. 

*  Since  there  are  36  segments  of  this  wheel,  or  a  =  10°  and      =  5°,  it  is  here  assumed  thai  ihe  length  of  ihe 

radius  is  the  distance  from  the  centre  to  the  joint  and  also  to  the  middle  of  the  rim-segment.  In  other  words,  it  is 
assumed  cos  5°  =  i,  which  involves  an  error  of  |  of  one  per  cent. 


BRIDGE-TRUSSES— ANALYSIS  FOR  WHEEL-LOADS. 


CHAPTER  V. 

ANALYSIS  OF  BRIDGE-TRUSSES  FOR  WHEEL-LOADS. 

87.  The  preceding  chapter  has  treated  all  live  loads  as  uniformly  distributed  While 
this  method  of  treatment  is  in  general  use  for  highway  bridges  and  to  some  extent  for  railway 
bridges,  it  has  become  the  general  practice  in  the  latter  case  to  deal  with  actual  specified 
wheel-loads  and  to  find  the  maximum  stress  in  each  member  due  to  these  loads.  In  highway 
bridges  also,  the  concentrated  load  specified  usually  determines  the  maximum  stresses  m  the 
floor-stringers  and  sometimes  in  the  floor-beams.  It  is  proposed  in  the  following  discussion 
to  show  the  method  of  finding  the  position  of  any  given  system  of  wheel-loads  which  will 
give  the  maximum  stress  in  any  member,  and  also  how  to  find  such  stress. 

DERIVATION  OF  FORMUL/E. 
PLATE  GIRDERS.     TRUSSES  W  ITH  PARALLEL  CHORDS  AND  VERTICAL  WEB  BRACING 

88.  Influence  Lines;  Definition.* — A  curve  representing  the  variation  of  moment, 
shear,  panel  load,  stress,  or  any  similar  function,  at  a  particular  point  in  a  structure  or  in  any 
particular  member,  due  to  a  load  unity  moving  over  the  structure,  is  called  an  influence  line 
The  difference  between  an  influence  line  and  an  ordinary  moment  or  shear  curve  is  that  the 
former  represents  the  variation  in  the  function  for  a  particular  point,  due  to  a  moving  load, 
while  the  latter  represents  the  variation  in  the  function  along  the  structure  due  to  some  fixed 
load. 

The  equation  of  the  influence  line  for  any  function  is  derived  by  writing  out  the  value  of 
the  function  for  a  load  unity  when  placed  at  a  variable  distance  x  from  one  end  of  the 
structure  taken  as  the  origin.  In  all  the  cases  here  treated,  the  equation  is  of  the  first  degree 
and  the  influence  lines  are  therefore  all  straight  lines. 

The  chief  use  of  influence  lines  is  in  determining  that  position  of  a  given  set  of  loads 
which  will  produce  the  maximum  value  of  any  function  ;  and  in  representing  to  the  eye  the 
influence  exerted  upon  the  value  of  this  function  by  the  various  elements  of  the  load,  and 
also  the  effect  of  shifting  the  loads  in  either  direction.  A  subordinate  use,  however,  of  these 
lines  is  in  getting  the  actual  value  of  these  functions. 


*  For  a  more  detailed  discussion  of  influence  lines  than  here  given,  see  a  paper  on  Stresses  in  Bridges  for  Con- 
centrated Loads  by  Prof.  G.  F.  Swain,  Trans.  Am.  Soc.  C.  E.,  July,  1887,  from  which  much  has  been  drawn  in  the 
following  discussion. 


74 


MODERN  FRAMED  STRUCTURES 


89.  Influence  Line  for  Bending  Moment  in  a  Beam,  or  at  any  Joint  of  the  Loaded 

Chord  of  a  Truss. — Let  C,  Fig.  122,  be  the  point  at  a  fixed  distance  a  from  the  left  end  ; 

and  let  x  =  the  distance  from  the  load  unity  to  the 

right  end.    Then  we  have,  when  P  is  to  the  right  of  C, 

>3  ^  ax , 

moment  at   C  —  P  j  X  a  =  j{P^  or  0-    This  is  the 

equation  of  the  straight  line  B'D,  where  the  ordinate 


C'D  = 


X  I.    Likewise  when  P  is  to  the  left 


of  C  the  moment  at  C  is  represented  by  ordinates  to 
the  line  A'D.  Then  A' DB'  is  the  influence  line  for 
moment  at  C. 

The  moment  at  C  due  to  a  load  P,  ,  at  any  point 
O,  is  equal  to  the  ordinate  y,  under  the  load,  multiplied  by  the  load;  for  the  ordinate  y  is 
equal  to  the  moment  due  to  unity  load.  The  moment  at  C  due  to  any  number  of  loads  may 
thus  be  found  by  multiplying  each  load  by  its  corresponding  ordinate  and  adding  the  several 
products. 

The  moment  at  C  due  to  any  length  of  uniform  load  of  p  per  unit  length  is  equal  to  the 
area  between  the  extreme  ordinates,  multiplied  by  p.  For  the  moment  due  to  an  element, 
pdx,  oi  load,  = />^/;i-j)/,  where  j;/ is  the  ordinate  under  the  load  ;  and  integrating  between  the 

limiting  values  of  x,  x^  and  x^  ,vve  have  :  total  moment 

moment  due  to  a  full  uniform  load  =  area  A' B' D  X  p- 

90.  Position  of  Moving  Loads  for  a  Maximum  Bending  Moment  in  a  Beam  or 
at  any  Joint  of  the  Loaded  Chord  of  a  Truss.— Let  A'DB',  Fig.  123,  be  the  influence 
line  for  moment  at  C. 

The  maximum  moment  due  to  a  uniform  load  is  when  the  load  extends  from  A  to  B,  and 

is  equal  to  the  area  A'DB'  multiplied  by  p,  where 
P  =  load  per  unit  length. 

—  a) 


p  lydx  =  /  X  area  EFHK.  The 


<  a  

C  1 

D 

1        ■     "  1 
1  1 

 j<  c2-j — ^ 

ail— a)  "-^ — 

Area  A  'DB'  X 


^a{l  —  a). 


Whence  the  maximum  moment  =  —a{l  —  aj.     .  (1) 


Fig.  123. 


Since  the  moment  at  C  due  to  any  load  P,  =  Py, 
this  moment  is  a  maximum  when  the  load  is  at  C  and 
^a{/  —  a) 


equal  to  P- 


l 


For  two  equal  loads,  P,  a  fixed  distance,  d,  apart,  the  rrfoment  is  evidently  a  maximum 
when  one  load  is  at  the  point  and  the  other  is  on  the  longer  segment  of  the  beam,  for  theo 
7  -|-/'  is  a  maximum.    This  maximum  moment  is  equal  to 


\     I        '        /  I  —  a  I 


2a(l  —  a)  —  ad 


(2) 


The  above  results  for  these  three  special  loadings  have  been  derived  in  Chap  TV,  though 
in  a  diff'erent  manner. 

The  position  of  any  given  system  of  loads  to  produce  a  maximum  moment  at  {Twill  now 
be  derived.    This  general  case  includes  also  the  above  special  cases. 

Let  G^,  Fig.  123,  represent  the  sum  of  all  the  loads  between  A  and  C  for  any  particular 


BRIDGE-TRUSSES— ANALYSIS  FOR  WHEEL-LOADS. 


75 


position  of  the  loading,  and  let  this  single  force  be  applied  at  the  centre  of  gravity  of  these 
loads.  In  a  similar  manner  let  replace  the  loads  on  the  right  of  C.  Now  since  G^  has 
the  same  ef?ect  upon  the  abutment  reaction  at  B  as  do  the  loads  which  it  replaces,  it  follows 
that  it  has  the  same  effect  upon  the  moment  at  C  as  do  these  loads.  Likewise  for  G^ 
Hence  the  moment  at  C  due  to  our  given  system  is 

M^G,y,-\-G,y,. 

Now  let  the  entire  system  move  a  small  distance  6x  to  the  left,  no  load  passing  A,  C,  or  B 
The  moment  will  become 

M-\-6M=  Gly,  -  6x  tan  a.)  +  G^y^  +  6x  tan  a^).^^ 

By  subtraction  we  have  '^^^^^    '        >-""3  V^/^^ 

8M=  G,8x  tan  -  G,6x  tan  a. ,          //        .    ^  .KfU^.-rd..  ^^^U— 

and  hence  U  ^^^^^^^"^'^'^ 

^               ^  (  G,         G,  \  . 

^—  =      tan  a,  -  G,  tan  =  '^'^\c^''  ~  A^'J ^3) 

For  a  maximum  value  of  M,        or  —  ^/^/  must  change  from  positive  to  nega- 

tive, that  is,  must  pass  through  zero,  as  the  loads  are  moved  to  the  left.  The  values  of  G^ 
and  G,  can  change  only  when  a  load  passes  A,  C,  or  B.  A  load  passing  A  decreases  C,  and  a 
load  passing  B  increases  G^,  but  a  load  passing  C  increases  G,  and  decreases  G^ ;  therefore  it 

G  G 

is  only  by  this  last  method  that  -pr^,  -rr^,  can  be  changed  from  positive  to  negative  and 

C  r>  AC 

the  moment  made  a  maximum.    For  a  maximum  moment,  then,  a  load  should  be  at  C  such 

G  G 

that  when  considered  as  part  of  G^  the  expression  ^i^>  —  j^>q  i  is  positive,  and  when  con- 

Q 

sidered  as  part  of      this  expression  is  negative.    Or,  stated  in  another  way,  when   .  '  ,  can 

A  C 

Q 

be  made  equal  to  ^t^i  by  considering  a  part  of  the  load  as  located  on  one  side  of  C  and  the 

remaining  part  as  located  on  the  other  side.    If  a  distributed  load  is  passing  C  this  condition 
can  be  definitely  satisfied. 

„  1  1-  ,  >  •  •  ^3  H~  ^1  ^1 

i^rom  the    equality  ^rr^/  =  A'C         have,  by  composition,     ^, ^,    =  ~^r^'i  ■>  " 
=     -f-  6^, ,  we  have 

-  U\ 
A'C'~  A'B' 

This  is  the  most  convenient  form  of  the  criterion  for  maximum  moment.  Expressed  in 
words  it  is  that  the  average  unit  load  on  the  left  of  the  point  must  be  equal  to  the  average  unit 
load  on  the  whole  span.  The  unit  of  length  may  be  taken  as  a  foot  or  a  panel  length.  The 
latter  is  the  more  convenient  for  trusses  of  equal  panels. 

For  a  given  set  of  wheel-loads  there  are  usually  two  or  more  positions  which  will  satisfy 
this  criterion.  The  moment  for  each  position  must  be  computed  and  the  greatest  value 
taken.  | 

91.  The  Point  of  Maximum  Moment  in  a  Beam  or  Plate  Girder. — Under  any  given 
position  of  the  loads  the  point  of  maximum  moment  is  the  point  of  zero  shear,  as  proved  in 
Mechanics,*  but  as  the  loads  are  shifted,  this  point  moves  and  the  moment  at  the  point 


See  also  Chap.  VIII. 


I 


76 


MODERN  FRAMED  STRUCTURES. 


varies.  The  problem  is  to  find  that  point  in  the  beam  at  which  this  moment  is  the  greatest 
that  can  occur  in  the  beam,  and  the  corresponding  position  of  the  loads. 

Let  G,  Fig.  124,  represent  the  sum  of  the  loads 
on  the  beam  AB\  b,  the  distance  from  B  to  their  cen- 
tre of  gravity ;  C,  the  required  point  of  maximum 
moment ;  and  ,  the  sum  of  the  loads  to  the  left  of  C. 
Then  the  condition  of  zero  shear  requires  that 


J: 


<  a— 

Fig.  124 


or    y  = 


If  the  moment  at  C  is  the  greatest  which  can  occur  in  the  beam,  then  it^^ust  be  the 
greatest  which  can  occur  at  the  point  C.    But  for  a  maximum  moment  at  any  point  we  have, 

G,_G 

a  ~  r 


by  eq.  (4), 

Hence  for  the  point  C 


G  G,       ,  , 

-f  =  -r  z=  — ,  whence  a  —  o. 
I       0  a 


(5) 


H — cJi-^ 

!  D 

1 

B' 

B 

\   D 

D 

/aids 

^  : 

b" 

«  dr-- 

^— d-2 — > 

Fig.  125 


Fig.  126 


The  point  C  will  lie  near  the  centre  of  the  beam  and  under  some  wheel  when  that  wheel 
and  the  centre  of  gravity  of  the  loads  are  equidistant  from  A  and  B.  This  condition,  together 
with  eq.  (4),  will  serve  to  determine  this  point  of  maximum  moment. 

92.  Influence  Line  and  Position  of  Loads  for  Maximum  Floor-beam  Concentration. 
— Let  AB  and  BC,  Fig.  125,  be  two  consecutive  panels  of  any  lengths     and  d^.    A  load  of 
unity  moving  from  C  to  B  produces  a  load  at 
B  proportional  to  the  distance  of  the  load  from 
C.    The  line  CD,  with  ordinate  DB'  equal 
unity,  is  then  the  influence  line  for  floor-beam 
load  at  B  when  the  loads  are  between  C  and  B- 
Likewise  DA'  is  the  influence  line  for  loads  on 
the  portion  BA.    The  load  at  B,  produced  by 
any  load  P,  is  equal  to  Py';  and  the  load  pro- 
duced by  a  uniform  load  p  per  unit  length  ex" 
tending  from  A  to  C  is,  equal  to 

P  X  area  A' DC  =  p  X  ^'      ^' ,  a  result  evident  by  inspection. 

The  influence  line  in  Fig.  125  is  of  the  same  for7}i  as  that  in  Fig.  123,  and  as  the 
criterion  for  maximum  moment  in  no  way  involves  the  length  of  the  ordinate  B'D,  the  same 
criterion  must  hold  good  for  maximum  panel  reaction.  That  is,  the  average  unit  loads  oh  the 
two  panels  must  be  equal  to  each  other  and  to  the  average  unit  load  on  both.  If  the  panels  are 
equal,  then  the  total  loads  in  the  two  panels  must  be  equal. 

If  A"D"C"  (Fig.  126)  be  the  influence  line  for  moment  at  ^  in  a  heamj^,  oi  length 

d,d^  —-"^^^ 
I,  =  d^-\-  d^,  the  ordinate  D"  B"  will  be  equal  to  ,  "i^.        the  ratio  of  any  two  correspond- 

ing  ordinates  y'  and  y"  in  Figs.  125  and  126  is  equal  to  the  ratio  of  DB'  to  D"B"  = 

I       f\  ' ,  =   '        .    Whence  the  floor-beam  reaction  at  B  in  Fig.  125  due  to  any  load  P  is 

<  +  < 

Moreover, 


d^  -\-  d^  d^d^ 

equal  to  the  moment  at  B,  Fig.  126,  due  to  the  same  load,  multiplied  by 


d,d„ 


since  the  maximum  floor-beam  concentration  occurs  for  the  same  position  of  the  loads  as  does 
the  maximum  moment,  we  have  only  to  find  the  maximum  moment  in  a  beam  of  length  equal 

to  d.^-\-  d^  at  a  distance  d^^  from  one  end  and  multiply  this  moment  by    '  T^^'  and  we  shall 


BRIDGE-TRUSSES— ANALYSIS  FOR  WHEEL-LOADS. 


77 


have  the  maximum  floor-beam  concentration  for  panel  lengths  of      and  d^.    If  d^=-  d^=  d. 


the  factor  is  -7. 

a 

93.  Influence  Line  and  Position  of  Loads  for  Maximum  Shear  at  any  Point  in  a 
Beam. — The  shear  at  C  in  the  beam  AB,  Fig.  127, 
due  to  a  load  unity  moving  from  B  towards  A  in- 


creases from  zero  for  the  load  at  ^  to  -|- 


for  the 


load  just  to  the  right  of  C.    As  the  load  passes  C  the 


shear  becomes  —  -j  and  increases  to  zero  at  A. 


Any 


movement  of  loads  to  the  left,  therefore,  increases  the 
positive  shear  at  C  until  some  load  P  passes  C,  when 
the  shear  is  suddenly  decreased  by  an  amount  equal  to 

^  -j^     7)  —        For  concentrated  loads,  therefore,  the  shear  reaches  a  maximum  each 

time  a  load  reaches  C.  The  greatest  of  these  maximum  values  evidently  occurs  when  one  of 
the  wheels  near  the  head  of  the  train  is  just  to  the  right  of  C. 

To  determine  which  of  two  consecutive  wheels,  P^  and  P^ ,  causes  the  greater  shear  when 
placed  just  to  the  right  of  C,  let  d  be  the  distance  between  these  wheels,  and  G'  the  total  load 
on  the  beam  when  /*,  is  at  C.  Let  the  loads  advance  a  distance  d  from  this  position,  thus 
bringing  P,  at  C.    The  effect  upon  the  shear  is  first  to  decrease  it  suddenly  by  an  amount  , 

G'l? 

and  then  to  increase  it  gradually  by  an  amount  equal  to  G  d  tan  a,  —  The  total  increase 

G'd 

is  therefore   P, .    According  as  this  expression  is  negative  or  positive  will  wheels  P^  or 

Pj  give  the  greater  shear.    For  equal  shear, 

G^_P, 

i-y  • 

The  slight  increase  in  shear  due  to  additional  loads  that  may  come  upon  the  beam  from 
the  right  has  been  neglected.    If  6^"be  the  total  load  on  the  beam  when  /*,  is  at  C,  then  the 

G'b  G"b 

increase  in  shear  will  be  somewhere  between  —j-  —  /*,  and  — ^  P,.    When  the  former 

expression  is  negative  and  the  latter  positive,  then  both  positions  should  be  tried.  This  will 
occur  only  for  a  short  distance,  to  the  left  of  which  both  arc  positive  and  to  the  right  both 
are  negative. 

94.  Influence  Line  and  Position  of  Loads  for  Maximum  Shear  in  any  Panel  of  a 
Truss. — Let  AB,  Fig.  128,  be  any  truss  and  FC  any  panel.    Let  n  =  total  number  of  pan- 
els, m  —  number  on  the  left  of  L\  and  vi'  tiie 


(6) 


number  on  the  ricfht.    Then  m' -. 


in.  The 


7  n 

 n 

/  a«  c' 

shear  in  FC  for  a  load  unity  moving  from  B  to 

C  will  be  equal  to  the  left  abutment  reaction 

1     Ml  ■  r  in'  , 

and  will  increase  from  zero  at  B  to  —  at  6. 

ti 

For  the  same  load  moving  from  F  to  A  the 
shear  is  negative  and  equal  to  the  right  abut- 
ment reaction,  thus  increasing  from  a  value 
m  —  I 

—  —  at     to  zero  at  A.    Between  T  and  F 


Fig.  128. 


the  load  is  carried  to  tliese  two  points  by  the 
floor-stringer,  the  amount  transferred  to  each  being  proportional  to  the  distance  of  the  load 


78 


MODERN  FRAMED  STRUCTURES. 


from  the  other.  The  shear  thus  varies  uniformly  and  the  influence  line  for  this  portioh  is 
the  straight  line  DK,  giving  the  complete  influence  line  A'KDB'. 

For  a  maximum  positive  shear  due  to  a  single  load  the  load  should  evidently  be  placed 
at  C.    A  uniform  load  will  give  the  maximum  positive  shear  when  extending  from  B'  to 
and  the  shear  will  then  be  equal  to  pX  area  EDB'.    Now  EC  :  DC'::  F'C  :  DC  +  F'K. 

whence  EC  =      X  d  X  —y-r-  =  '—-    a  result  already  found  in  Art.  60.  The 

n  m  -f-  7/1  —  \      ji  —  x'  ■'  ^ 

shear  =p  X  area  EDB'  -  p  x  ^~  X  ('«'^+  =  ^^^(^^"1.  same  as  eq.  17  (a), 

page  51. 

For  concentrated  loads,  let  G,  represent  the  portion  from  A  to  F,  G,  that  in  the  panel 
FC,  and  G^  that  on  the  right  of  C.    We  then  have  the  total  positive  shear 

Let  the  loads  move  a  small  distance  6x  to  the  left.    The  shear  will  then  be 

S-\-  SS=  G^y,  +  Sx  tan  a-,)  -]-  GJ^y,  —  6x  tan  a,)  —  G\{y^  —  dx  tan  of,). 
By  subtraction  and  division  we  have 

-^  —  G^  tan  n-,  —  G^  tan  «,  +      tan  a, 

XT  ■  ^ 

1^  or  a  maximum, —  =  o  ;  whence 
ox 

G,  -  G,{n  -  i)  +     =  o, 

or 

That  is,  /or  a  maximum  shear  in  avy  panel  the  load  in  the  panel  must  be  equal  to  the  load  on  the 

SS 

truss  divided  by  the  number  of  panels.    The  only  way  ^  can  be  made  zero  by  passing  from 

positive  to  negative,  and  so  giving  a  maximum  and  not  a  minimum  as  the  loads  are  moved  to 
the  left,  is  by  a  load  passing  C  or  A.  Rut  for  the  greatest  positive  shear  there  should  as  a  rule 
be  no  loads  on  the  portion  AF.  Hence  the  maximum  shear  will  occur  with  some  wheel  near 
the  he.ad  of  the  train  at  C,  such  as  will  satisfy  the  above  criterion. 

Since  a  wheel  passing  C  causes  a  large  relative  increase  in  (j, ,  it  will  be  found  that  the 
maximum  shears  in  several  panels  will  occur  with  the  same  wheel  at  the  pane'  point  to  the 
right.  To  find  for  what  panels  any  particular  wheel  placed  at  the  panel  point  to  the  right 
will  give  a  maximum  shear,  let  G^  represent  the  wheels  in  the  panel  other  than  the  wheel  P. 
Then  if  /"at  the  right-hand  panel  point  gives  a  maximum,  the  criterion  requires  that  G  >  nG, 
and  <  n{G,  -j-  P).  Wheel  /"at  the  right-hand  panel  point  will  then  give  a  maximum  so  long 
as  the  entire  load  upon  the  bridge  lies  between  nG.,  and  n^G,  -|-  P). 

COMPUTATION  OF  STRESSES.     PLATE  GIRDERS.     TRUSSES  WITH  PARALLEL  CHORDS  AND 

VERTICAL  WEB  BRACING. 

95.  Wheel-loads;  Tabulation  of  Moments. — A  diagram  such  as  is  shown  in  Fig.  129 
is  of  great  assistance  in  finding  stresses  due  to  actual  wheel-loads.*  The  diagram  in  the  figure 
*  See  p.  108^  for  a  similar  moment  table  for  Cooper's  Conventional  Loads. 


BRIDGE- 


TRUSSES— ANALYSTS  FOR  WHEEL-LOADS. 


79 


t  <D  w  o  lO  in  . 


CO  S 


z 
111 


>• 
< 
llJ 
I' 


tiJ 


CO 
CO 


o 

CO  oi 

o  ° 

13 

2  § 
2  5 


i3 


is  for  Cooper's  Class  "  Extra  Heavy  A."  Such  a 
diagram  should  be  made  up  for  each  system  of  loads 
in  use. 

All  loads  are  given  in  thousands  of  pounds  and 
all  moments  in  thousands  of  foot-pounds.  The 
loads  given  are  one  half  the  total  loads,  being  those  on 
one  rail.  All  distances  are  given  in  feet  and  tenths, 
instead  of  feet  and  inches  as  in  Cooper's  specifications. 

The  various  horizontal  columns  contain  the 
following  data,  beginning  at  the  top  : 

1.  Summation  of  loads  from  right  end. 

2.  Summation  of  loads  from  left  end. 

3.  Distance  between  wheels. 

4.  Summation  of  distances  from  left  end. 

5.  Summation  of  distances  from  right  end. 

6  to  15.  Summation  of  moments  about  the  wheel 
vertically  over  the  step  in  the  heavy  stopped  line.  . 

The  weight  on  each  w^heel  is  given  in  heavj- 
figures  on  the  respective  wheels  and  is  in  thousands 
of  pounds.  The  figure  in  the  small  circle  is  the 
number  of  the  wheel  from  the  left  end. 

96.  Application  of  the  Foregoing  Principles 
to  a  Pratt  Truss. — Let  us  take  a  truss  of  200  ft. 
.span  with  eight  panels  and  a  height  of  32  ft.,  Fig. 
130.    Live  load  only  will  be  considered  here. 

Chord  Stresses. 

The  maximum  moments  at  b,  r,  d,  and  r  di- 
vided by  32  ft.  will  give  us  the  maximum  chord 


Fig.  130. 

stresses.  These  maximum  moments  will  now  be 
found  by  the  aid  of  the  diagram  Fig.  129.  The 
criterion, for  maximum  moment  is  that  the  average 
load  per  unit  length  on  the  left  must  equal  the 
average  load  per  unit  length  on  the  whole  bridge. 
A  panel  length  will  be  taken  as  the  unit. 

1st.  Moment  at  b. — It  is  evident  that,  in  gen- 
eral, the  bridge  should  be  loaded  with  the  heaviest 
possible  load,  and  that  heavy  concentrations  will 
have  the  most  effect  when  near  the  point  of  mo- 
ments. 

Let  us  try  wheel  2  at  b.  The  bridge  will 
extend  175  ft.  to  the  right  of  2,  and  hence 
175  —  94.9,  =  80. 1  ft.,  of  uniform  load  will  be  on 
the  bridge.     Total  load  then  =l  G  —  208  -|-  80.1 


1  o  o  ej 


S2  S  59 


8o  MODERN  FRAMED  STRUCTURES. 

X  1.5  —  328.1,  and  — .  =  — —  41.    Load  to  the  left  divided  hy  ai>,  =  -j^  =  —  ,  =  8,  con 

M  o  (10  I 

sidering  wheel  2  as  on  the  right ;  or  =  23,  considering  wheel  2  as  on  the  left.    Since  both  these 

Q 

values  of  are  less  than  ^ ,  the  moment  at  b  will  be  increased  by  moving  the  loads  to  the 
left. 

Try  wheel  3  at  b. 

G     208  +(175-89.1)1.5 

 s  ■  =  42.1, 

at  o 

23^  38 

— r  =  —  to  — , 

ab      I  I 

and  hence  the  moment  will  be  increased  by  moving  the  loads  to  the  left. 
Try  wheel  4  at  b. 

G     208 +  (175  -  84.6)1.5. 

=   _  42.9, 

at  o 

38^  53 
— 7  =  —  to  — . 
ab      I  I 

Q 

One  of  these  values  being  less  and  the  other  greater  than  — .  ,  wheel  4  at  b  gives  a-maximum 
moment. 

To  find  this  moment,  get  first  the  abutment  reaction  at  a.  Line  6  gives  the  moment  of 
the  wheels  about  the  point  19  at  the  left  end  of  the  uniform  train  load.  It  is  11 233.  The  right 
abutment  is  175  —  84.6,  —  90.4  ft.,  to  the  right  of  19.    Hence  the  total  moment  of  the  load 

1.5  X  (90.4)' 

about  the  right  abutment  is  11233  +  208  X  90.4  +  — —  =  36168.     Left  abutment 

36168  ,  ,  ,  .       ,        ,       36168  36168  ^. 

reaction  =  .    Moment  of  left  reaction  about  0  =  7:;   x  25  =  — - —  =  4521.  The 

8  X  25  8  X  25  ^    ^  8 

moment  of  the  wheel-loads  to  the  left  of  b  about  b  is  given  in  line  12  between  wheels  10  and 

II,  and  is  to  be  subtracted  from  the  moment  of  the  abutment  reaction.    Hence,  moment  at 

b  =  4521  —  369  =  4152,  thousand  ft.-lbs. 

Let  us  see  if  there  are  other  maximum  values.    With  wheel  5  at  ^,  —  =  43.8  and 

8 

G,  —  53  to  68,  thus  showing  that  this  position  does  not  give  a  maximum.  As  the  loads  are 
moved  farther  to  the  left,  G^  will  be  made  less  than  —  by  some  wheel  passing  a,  but  this  causes 

o 

a  minimum,  not  a  maximum.    As  the  loads  are  moved  still  farther,  G^  will  again  be  made 

greater  than  --  by  some  heavy  wheel  11,  12,  or  13  passing  b.    This  will  give  other  maxima, 
o 

but  not  so  great  as  that  found  above,  since  now  we  have  a  uniform  train-load  on  the  right,  in 
place  of  heavy  engine-loads.    The  actual  moments  for  wheels  il,  12  and  13  at  b  are  3802, 
3923,  and  3801,  respectively.    Wheel  4  at  b  therefore  gives  the  greatest  possible  moment. 
2d.  Moment  at  c. — Try  wheel  6  at  c. 

G     208 +  (150 -73)1.5 

ai  8  =  4^-4' 

G     68  77 

— -  =  —  to  —  =  34  to  38.    Hence  no  maximum. 
ac      2  2 


BRIDGE-TRUSSES— ANALYSIS  FOR  WHEEL-LOADS. 


8i 


Wheel  7  at  c. 


G     208 -1- (150-68.2)1.5 


at 


77  86 
- '  =  f-^  to  —  =  38  to  43. 
ac      2  2 


-  41.3. 


Hence  wheel  7  gives  a  maximum  moment  at  c.  Wheel  8  i.s  found  to  give  no  maximum,  and 
hence  wheel  7  gives  the  greatest  possible  moment. 


1 1233  4-  208  X  81.8  + 


1.5  X  (81.8/ 


Moment  at  c 


X  2  —  1460. 1  =  6857.4. 


3d.  Moment  at  d. — Wheels  11  and  12  are  found  to  give  maximum  moments.  The  corre- 
sponding moments  are  8530.5  and  8536.0. 

4th.  Moment  at  e. — Wheels  13  and  14  at  e  give  maxima.  The  corresponding  moments  are 
9029.0  and  9030.9. 

We  have,  then,  by  dividing  the  moments  by  32,  the  following 

LIVE  LOAD  CHORD  STRESSES. 


Point. 

Max.  Moment. 

Pieces. 

Stresses. 

b 

4152 

ahc 

129.8 

c 

6857 

cd  and  BC 

214-3 

d 

8536 

de  and  CD 

266.8 

c 

9031 

DE 

282.2 

Web  Stresses. 

The  maximum  stresses  in  all  the  web  members  except  bB  and  JiH  result  directly  from 
the  maximum  shears  in  the  various  panels.  Those  in  bB  and  hH  a.Ye  equal  to  the  maximum 
panel  load,  which  will  be  found  below.  The  criterion  for  maximum  shear  is  that  the  total  load 
divided  by  the  total  number  of  panels  must  equal  the  load  in  the  panel. 

1st.  Shear  in  ab. — For  this  panel  the  criterion  for  shear  is  the  same  as  for  moment  at  b, 
hence  the  same  position  will  give  a  maximum  in  both  cases.  Wheel  4  is  at  b.  Shear  —  abut- 
ment reaction  minus  panel  load  at  a.   This  abutment  reaction  was  found  to  be  —  180.8. 

^  8  X  25 

The  panel  load  at  a  is  obtained  by  finding  the  moment  of  the  wheels  i,  2,  and  3  about  b  and 
dividing  by  the  panel  length.    This  moment  is  given  in  line  12  between  wheels  10  and  li  and 

369.2 


is  equal  to  369.2.    Whence  shear  =  180.8 


!5 


—  166.0. 


2d.  Shear  in  be. — Try  wheel  3  at  (.   Total  load  on  the  bridge  =  G=  208  -)-(i50  —  89.1)1.5 

Q 

—  299.3.        —  37.4.    The  load  in  the  panel  =  G^,  =  23  or  38.    Hence  wheel  3  at  <:  gives  a 

o 

maximum.    Shear     abutment  reaction  minus  load  at  b 

1.5  X  (60.9)" 


1 1236.2  4-  208  X  60.9  -|- 


200 


198.2 
25 


1334 -7-9=  125.5. 


82 


MODERN  FRAMED  STRUCTURES. 


Try  wheel  4  at  c. 


G_  208  4- (150—  84.6)1.5 

8  ~  8 


=-  38.3, 


G,=  Z?>  to  53 
1 1236.2  +  208  X  65.4  + 


Shear  = 


Hence  wheel  4  gives  a  maximum. 

1-5  X  (65.4)' 

2  369.2 


200 


25 


=  125.4. 


As  the  loads  are  moved  still  farther  to  the  left,      becomes  greater  than  — ;  it  is  then 

8 

made  less  by  some  wheel  passing  b,  thus  giving  a  minimum,  and  is  again  made  greater  by 
some  wheel  il,  12,  or  13  passing  c,  thus  giving  another  maximum.  Such  a  maximum  is 
evidently  much  less  than  that  found  above. 

3d.  Shear  in  cd. — Wheel  3  at     gives  the  only  maximum  value.    Shear  =  90.4. 

4th.  Shear  in  de. — Wheel  3  at  r  gives  the  only  maximum,  the  value  of  which  is  60.I. 

5th.  Shear  in  ef. — Wheels  2  and  3  give  maxima.  With  wheel  2  at  /,  the  end  of  the  uni- 
form train-load  is  94.9  —  75,  =  19.9  ft.,  to  the  right  of  the  end  of  the  bridge.  Wheel  14  is 
therefore  the  last  wheel  to  the  right  that  is  on  the  bridge,  and  is  26.5  —  19.9,  =6.6  ft.,  to  the 
left  of  i.    Moment  about  i=  moment  about  wheel  14 -f-  172  X  6.6.  Hence 


With  wheel  3  at  /, 


shear  in  ef  ■ 


shear  m  ef  ■= 


6257.0+  172  X  6.6  64.8 
200  25 


=  34-4. 


8347.0 -f  190  X  0.5  _  198.2 

200  25 


34-3- 


6th.  Shear  in  fg. — Wheel  2  at  ^  gives  a  maximnm,  whose  value  is  16.27.  This  positive 
shear  would  probably  be  less  than  the  dead  load  negative  shear  and  would  therefore  be  needed 
only  to  show  that  fact. 

The  dead  load  negative  shear  to  the  right  of  ^  would  be  much  greater  than  the  live  load 
positive  shear. 

Multiplying  the  above  maximum  shears  by  cosec  d,  =■  25'  -(-  32'  32,  =  1.269,  for  the 
stresses  in  the  diagonals,  we  have  the  following 

LIVE-LOAD  WEB  STRESSES. 


Panel. 

Shear. 

Piece. 

Stress. 

Piece. 

Stress. 

ab 

166.0 

aB 

+  210.6 

be 

125-5 

Be 

-  159-3 

cd 

90.4 

cC 

4-90.4 

Cd 

-  II4-7 

de 

60.1 

dD 

-I- 60.1 

De 

-  76.3 

'/ 

34-4 

eE 

+  34-4 

Ef 

-  43-7 

The  proper  positions  of  the  loads  might  have  been  more  quickly  found  by  finding  for 
what  panels  each  of  the  wheels  2,  3,  and  4  would  give  a  maximum  shear  when  placed  at  the 
right-hand  panel  point,  as  explained  in  Art.  94.  Thus  with  wheel  3  at  this  panel  point,  G^ 
varies  from  23  to  38  and  8G^,  lies  between  184  and  304,  which  are  then  the  limiting  values  of 
G  for  wheel  3  to  give  a  maximum.  Wheel  3  therefore  gives  a  maximum  when  the  end  of  the 
bridge  lies  between  wheel  16  and  (304  ■—  208)  1.5,  =  64  ft.,  to  the  right  of  point  19;  that  is, 
when  wheel  3  itself  is  from  74.5  ft.  to  153.1  ft.  to  the  left  of  the  right  end  of  the  bridge. 
This  includes  panel  points  f,  e,  d,  and  c,  as  found  above. 


BRIDGE-TRUSSES— ANALYSIS  EOR  WHEEL-LOADS. 


83 


The  field  of  wheel  4  is  likewise  found  to  lie  between  148.6  ft.  from  the  right  end,  to  the 
left  end,  thus  including  points  c  and  b. 

The  field  of  wheel  2  is  from  14.8  ft.  to  80.3  ft.  from  the  right,  thus  including  points  h, 
g,  and  /. 

It  is  to  be  noticed  that  consecutive  fields  overlap  by  an  amount  equal  to  the  distance 
between  wheels. 

Floor-beam  Reaction  ;  Stresses  in  Hip  Verticals. 
These  are  a  maximum  when  the  joint-load  is  a  maximum.    By  Art.  92  the  maximum 
joint  load  is  equal  to  the  maximum  moment  at  the  centre  of  a  beam  50  feet  long,  multiplied 
2 

by->.    Wheel  4  at  the  centre  of  such  a  beam  is  the  only  wheel  giving  a  maximum  moment. 

(1958.9  +  95  X  2.81    =        ^^'^^  ^'""^^'^^  -  '^^^^^M^^^' 
The  value  of  this  moment  =  >  369.2  =  743-2.  Hence  the  maximum  fioor- 

2 

beam  load  =  743.2  X  —  =  59-46.    Two  such  loads  applied  on  the  floor-beam  at  the  points  of 
25 

attachment  of  the  stringers  will  produce  the  maximum  moment  and  shear  in  the  beam. 
These  are  readily  computed. 

In  the  diagram.  Fig.  129,  the  floor-beam  concentration  or  stress  in  hip  vertical  may  be  quici<ly  found  by 
placing  wheel  13  over  the  floor-beam  and  then  deducting  from  the  total  load  on  the  two  adjacent  panels  the 
part  whicii  is  transferred  to  the  two  adjacent  floor-beams  by  the  stringers.  Thus,  as  wheels  10  to  17  are 
on  the  two  panels,  the  total  load  is  95  and  the  part  transferred  to  the  adjacent  beams  is.  taking  moments 

about  wheel  13  and  which  are  given  in  line  12,^^^"^  ^  5'9-3  _      g_    The  beam  concentration  is  therefore 

95  -  35-5  =  59-5- 

Stringers  ;  Maxinmvi  Moment  ajid  Shear. 

The  maximum  moment  will  be  near  the  centre  and,  by  Art.  91,  will  be  under  some  wheel 
when  that  wheel  and  the  centre  of  gravity  of  the  entire  load  on  the  stringer  are  equally 
distant  from  the  centre.    Length  =  25  ft. 

It  will  be  simplest  to  first  find  what  wheels  placed  at  the  centre  will  give  a  maximum 
moment  at  that  point.    These  are  found  by  the  criterion  of  eq.  4,  p.  75,  to  be  wheels  3  and  4. 

With  wheel  3  at  the  centre,  uhecl.s  2  to  5  are  on  the  stringer.  The  distance  of  the  centre 
of  gravity  of  these  four  wheels  from  5  is  found  by  taking  moments  about  wheel  5  or,  as  given 
in  line  11  of  the  diagram,  about  wheel  14,  to  be  424.5  60  =  7.1  ft.  ;  hence  wheel  3  should  be 
placed  a  distance  (9  —  7.1)     2  =:  .95  ft.  to  the  left  of  the  centre.    The  moment  under  wheel 


II  1?  c 

3  is  then  found  to  be  233.5,  —  60  X  — '~  —  87. 

With  wheel  4  at  the  centre,  wheels  2  to  6  are  on,  and  the  centre  of  gravity  of  these 
five  loads  is  found  to  be  12.3  ft.  to  the  left  of  6,  and  .7  ft.  to  the  left  of  4.  With  wheel  4 
.35  ft.  to  the  right  of  the  centre  the  moment  under  4  is  found  to  be  234.7,  the  required 
maximum. 

The  moments  at  the  centre  under  wheels  3  and  4  are  230.2  and  234.3  respectively,  values 
very  nearly  as  great  as  the  above,  thus  showing  that  the  moment  varies  but  slightly  near  the 
centre  of  the  stringer. 

The  maximum  shear  in  the  stringer  occurs  at  the  end,  when  one  of  the  wheels  is  about 
to  pa.ss  off.  Wheel  2  at  the  left  end  will  evidently  give  a  greater  reaction  than  will  wheel  i. 
With  wheel  2  at  the  left  end,  wheel  6  will  be  (8.1  +  25)  —  30.O  —  3.1  ft.  to  the  left  of  the 

right  end.    Left  reaction  =  shear  =.  +  ^"^^^A  =  42.58.    With  wheel  3  at 

the  left  end  the  shear  is  found  to  be  41.58.    The  required  maximum  is  therefore  42.58. 

97.  Plate  and  Lattice  Girders. — The  maximum  moments  and  shears  are  found  at 
sc  .  oral  points  along  the  girder,  and  from  these  the  flange  and  web  stresses  are  obtained.  In 
addition  to  these  moments  the  maximum  moment  possible  in  the  girder  may  be  found  as  in 


84 


MODERN  FRAMED  STRUCTURES. 


the  case  of  the  floor-stringer  above,  by  testing  a  few  wheels  which  give  a  maximum  at  the 
centre. 

The  other  moments  are  found  by  the  same  method  as  illustrated  in  the  preceding  article. 
The  shear  at  any  point  is  a  maximum  with  either  wheel  i  or  2  just  to  the  right  of  the 

G'  P 

point.    Wheel  2  at  the  point  will  give  a  maximum  whenever  -^-5.       and  wheel  i  will  give  a 
G"  P 

maximum  when  where  G'  =  load  on  the  girder  with  wheel  i  at  the  point,  G"  =  load 

on  the  girder  with  wheel  2  at  the  point,  b  —  distance  between  wheels  i  and  2,  and  P^  —  weight 
of  wheel  i.    This  follows  from  Art.  93. 

Thus  for  a  60-foot  girder,  ^  =     -  =  .99  and  -^-/  =  59.4.    Hence  when  G'  >  59.4,  wheel  2 

gives  the  greater  shear;  that  is,  when  five  or  more  wheels  are  on  the  girder  with  wheel  i  at  the 
point,  which  would  be  for  all  points  22.9  ft.  or  more  from  the  right  end.  When  G"  ^  S9-4 
or  when  there  are  four  or  less  loads  on  the  girder  with  wheel  2  at  the  point,  then  wheel  i 
gives  the  greater  shear  ;  that  is,  for  all  points  less  than  14.8  feet  from  the  right  end.  Between 
14.8  and  22.9  feet  from  this  end  both  positions  should  be  tested. 

98.  Graphical  Methods.  Load  Line  and  Moment  Diagram. — The  following  graphical 
methods  of  computing  stresses  for  actual  wheel  loads  are  mostly  due  to  Dr.  Henry  T.  Eddy. 
They  are  given  by  him  in  a  very  elaborate  paper  discussing  the  whole  subject,  in  the  Trans. 
Am.  Soc.  C.  E.,  Vol.  XXII,  1890,  paper  No.  437. 

These  methods  offer  considerable  advantage  over  the  analytical  methods  as  regards  speed, 
and  if  the  diagrams  are  drawn  carefully  and  to  a  large  scale  they  are  sufficiently  accurate  for 
computmg  the  stresses  in  the  main  members  of  a  truss.  For  floor-beams  and  stringers  the 
moment  table  should  be  used  in  getting  moments  and  shears,  but  the  diagram  may  be  used 
in  getting  the  position  of  the  loads. 

The  diagram,  Fig.  131,  is  constructed  as  follows:  Upon  profile  paper,  or  upon 
specially  prepared  cross-section  paper,  is  first  laid  off  the  wheel-diagram,  to  a  horizontal  scale 
of  about  8  ft.  to  the  inch.  The  wheels  are  numbered,  their  distances  apart  noted,  and  vertical 
lines  drawn  through  each.  The  stepped  "  load  line,"  1-2-3-4-,  etc.,  is  simply  a  line  whose 
ordinates,  measured  from  the  horizontal  reference  line  o-o  at  the  bottom  of  the  sheet,  are 
equal  to  the  summation  of  the  loads  to  the  left.  It  thus  consists,  over  the  wheels,  of  a  series 
of  steps,  each  step  being  equal  by  scale  to  the  load  below.  Over  the  uniform  load  it  is  a 
straight  line  rising  at  the  rate  of  1500  lbs.  per  foot.  A  convenient  scale  for  loads  is  about 
20,000  lbs.  to  the  inch. 

The  moment  lines,*  numbered  i,  2,  3,  4,  .  .  .  18,  at  the  right  edge,  are  constructed  by 
beginning  at  the  horizontal  reference  line  at  some  point  near  the  right  end  of  the  sheet  and 
laying  off  successively  on  a  vertical  line  the  moment  of  each  wheel  about  that  point,  beginning 
with  wheel  I.  Then  the  line  i  is  drawn  from  the  reference  line  over  wheel  i  to  the  first  point 
thus  found  ;  the  line  2  through  the  intersection  of  line  i  with  the  vertical  over  wheel  2,  and 
the  second  point  so  found  ;  then  3  through  the  intersection  of  2  with  the  vertical  over  wheel 
3,  and  the  third  point ;  etc.  A  scale  of  one  one-hundredth  of  the  scale  for  loads  will  be  found 
convenient.  The  ordinate  at  any  point,  from  the  reference  line  to  any  one  of  these  moment 
lines,  is  thus  equal  to  the  sum  of  the  moments  about  the  point  of  the  loads  up  to  and  includ. 
ing  the  load  corresponding  to  the  moment  line  taken.  Also  the  ordinate  at  any  point  between 
any  two  lines,  as  between  2  and  8,  is  equal  to  the  sum  of  the  moments  of  wheels  3  to  8,  inclusive, 
about  that  point.  The  broken  line  AB,  formed  of  segments  of  moment  lines,  is  evidently  an 
equilibrium  polygon  for  the  given  loads.  The  portion  BC  above  5  is  a  parabolic  curve  and  can 
be  easily  constructed  by  computing  the  moments  of  the  portion  of  the  uniform  load  between  B 
and  points  to  the  right,  about  those  points,  and  laying  off  these  moments  above  the  line  18. 

*  This  part  of  the  diagram  is  due  to  Prof.  Ward  Baldwin,  M.  Am.  Soc.  C.  E.  See  Eng.  News,  Sept.  28,  Oct.  12, 
and  Dec.  28,  1889. 


ANALYSIS  FOR  BRIDGE-TRUSSES  FOR  WH EEL-LOADS. 


85 


99.  Application  of  the  Diagram,  Fig.  131,  to  Finding  Maximum  Moments.  -Before 
using  the  diagram,  the  panel  points  of  the  truss,  or  the  points  along  the  beam  where  the 


Feet  aol  I  I  llO 


Fir..  131. 


A.ai.BADl(  Not«  Co.N«r. 


moments  are  desired,  are  first  marked  off  on  a  slip  of  paper  to  the  same  scale  as  the  diagram. 
Suppose  1-2-3-4  .  .  .,  Fig.  132,  represent  a  portion  of  a  load  line,  and  A,  C,  D,  E,  B  be  the 
panel  points  of  a  truss,  marked  off  on  a  separate  slip  of  paper.    Let  it  be  required  to  find  the 

position  of  loads  for  a  maximum  moment  at  D. 

Q 

The  average  load  on  the  left,  —        ,  must  equal 

Q 

the  average  load  on  the  bridge,  —  —^^^ .  Try 
wheel  4  at  Z>  by  placing  D  under  this  wheel.  The 

Q 

ordinate  EB  =  G,  and  -^-^  is  the  slope  of  the  line 

Q 

AF.    The  ordinate  ID  or  KD  —  G,,  and 

is  equal  to  the  slope  of  AI  or  AK.  The  slope  of 
AFh  seen  to  be  greater  than  that  of  AK  and  less 
than  that  of  AT;  therefore  this  position  gives  a 
maximum  moment  at  D.  Hence,  in  general,  for 
a  maximum  moment,  if  the  line  whose  slope  is 


86 


Modern  framed  structures. 


Q 

equal  to  cuts  the  load  or  step  which  is  above  the  point,  the  position  is  one  giving  a  maxi- 
mum. This  line  is  conveniently  indicated  by  means  of  a  thread.  If  there  are  no  loads  off  the 
bridge  to  the  left,  the  line  starts  from  the  zero  line  at  the  end  of  the  bridge  at  A  ;  but  if 
there  are  loads  to  the  left,  it  starts  in  the  load  line  vertically  over  the  end  of  the  bridge. 
The  right  end  is  the  point  in  the  load  line  vertically  over  the  right  end  of  the  bridge.  In  the 
above  case,  if  y^/'"  should  pass  above  AI,  the  loads  must  be  moved  to  the  left;  and  if  it  should 
pass  below  AK,  then  the  loads  must  be  moved  to  the  right. 

The  moment  itself  is  easily  found  on  the  moment  diagram  by  reading  of?  the  ordinate  at 

AD 

B  between  the  moment  lines  o-o  and   7,  multiplying  this  by  -^-^ ,  and  subtracting  the 

moment  of  the  loads  to  the  left  about  D,  which  is  given  by  the  ordinate  at  D  between  0-0  and 
4.  The  moment  at  D  is  also  equal  to  the  ordinate  from  the  closing  line  to  the  equilibrium 
polygon  formed  by  the  segments  of  the  moment  lines.  The  extremities  of  this  closing  line 
lie  in  verticals  through  the  ends  of  the  bridge. 

In  finding  the  greatest  possible  moment  in  a  girder,  the  centre  of  gravity  of  any  number 
of  loads  is  readily  found  by  producing  the  two  segments  of  the  equilibrium  polygon  including 
these  loads,  to  their  intersection. 

100.  Application  of  the  Diagram,  Fig.  131,  to  Finding  Maximum  Shears. — ist.  S/iear 
in  Beams  or  Girders. — It  has  been  shown  in  Art.  93  and  illustrated  in  Art.  97  that  wheel  i 

G"  P 

will  cause  a  maximum  shear  at  all  points  to  the  right  of  the  point  where  —j-  =       and  that 

G'  P 

wheel  2  will  cause  a  maximum  shear  to  the  left  of  the  point  where  —  =  ~;  where  G'  and  G" 

are  the  total  loads  on  the  girder  with  wheels  i  and  2,  respectively,  at  the  point ;  b  —  distance 
between  wheels  i  and  2  ;  =  weight  of  wheel  i  ;  and  /  =  length  of  girder.  (Stringers  less 
than  about  18  ft.  long  are  an  exception,  they  having  a  maximum  end  shear  with  wheel  3  at 
the  end.) 

P 

Let  A\-2  ...  8,  Fig.  133,  be  a  load  line.    Draw  the  line  A  T,  having  a  slope  of  — .  Lay 

off  the  length  of  the  girder  AB  on  the  line  0-0,  and  at  B  draw  the  vertical  BT.    This  vertical 

P 

ordinate  is  that  load  which,  divided  by  /,  =     ,  and 

18 

[7 — '  therefore  is  the  load  G'  or  G"  above  mentioned.  In 

p  order  that  this  may  be  the  total  load  on  the  girder,  wheel 

5  must  be  at  the  right  end.    Laying  off  the  girder  then 

from  B'  to  the  left,  the  portion  B'C  to  the  right  of  wheel 

2  is  the  portion  of  the  girder  in  which  the  maximum 

shear  is  given  by  wheel  i,  and  the  portion  of  the  girder  to 

  o      the  left  of  C"  has  its  maximum  shears  for  wheel  2. 

°'  ^'  ^  Between  C'  and  C"  both  positions  should  be  tested. 

Fig.  133. 

The  line  ^  7"  is  permanently  drawn  on  the  diagram  ;  no 

other  lines  need  be  drawn. 

2d.  Shear  in  a  Truss. — The  criterion  for  maximum  shear  may  be  stated  thus:  the  total 

load  on  the  truss  divided  by  its  length  must  equal  the  load  in  the  panel  divided  by  the  panel 

length.    Any  wheel  /'at  the  panel  point  to  the  right  will  thus  give  a  maximum  shear  so  long 

G        ^  G^         G,  +  P 

as  y  hes  between       and  — ^ — ,  where  G  =  total  load  and  G,  =  load  in  the  panel  other 

than  P. 

Let  I  ...  15,  Fig.  134,  be  any  load  line.    Mark  off  on  a  strip  of  paper  the  panel  points 


BkIDGE-TRUSSI<:S— ANALYSIS  I- OR  W H EEL- LOADS. 


«7 


of  the  truss  and  place  it  in  the  position  AB,  with  the  firbt  panel  point  C  under  wheel  I. 
panel  points  are  D,  E,  and  F.  Call 
the  loads  P^,  P,,  P^,  etc.  Draw 
the  lines  AT^,  AT,,  AT,,  etc., 
having  ordinates  at  C  equal  to  P, , 
p^^P^,P^-{-P^-\-P^,etc.  These 
lines  will  then  have  the  slopes  of 
P,  +         P,+P.  +  P. 

d  ' 


Other 


etc., 


d     '  d 
respectively. 

Wheel  I  at  the  right-hand 
panel  point  of  any  panel  will  give  a 

maximum  shear  so  long  as  y  <.  "^' 

This  limiting  value  of  G  is  the 
ordinate  BT^  at  the  right  end  of 
the  truss.  Hence  /*,  gives  a  maxi- 
mum until  P^  comes  on,  that  is,  for 
the  distance  B' C  at  the  right  end 


r 

/ 

/ 

/ 

Jl4 

/  1— 
/   |l3 

 ^12 

In 

DB 


or  for  values 


of  the  truss.    Wheel  2  gives  a  maximum  for  values  of  -7  between  — ^  and  , 
°  lad 

of  G  between  BT^  and  BT^.    That  is,  wheel  2  gives  a  maximum  shear  in  the  panel  to  the  left 

of  each  joint  crossed  by  it  in  moving  the  loads  from  the  position  with  wheel  5  at  ^  to  the 

position  with  wheel  10  at  B.    The  strip  of  paper  should  be  placed  with  the  point  B  first  at  B' 

and  then  at  B" ,  the  point  below  wheel  2  being  marked  in  each  position.    The  space  between 

these  marks  is  the  space  in  which  the  shear  is  dominated  by  P^.    It  is  seen  to  overlap  the 

space  B' C,  dominated  by  wheel  I,  by  the  distance  between  wheels,  as  shown  in  Art.  96.  The 

ordinate  from  B  to  A     being  greater  than  G  when  P,  is  at      the  last  panel  point,  we  may 

say  that  P,  dominates  the  shear  in  the  truss  from  a  position  with  wheel  10  at  B,  to  the  left 

end.    We  thus  have,  very  briefly,  the  position  for  maximum  shear  in  all  the  panels.    None  of 

the  lines  need  be  actually  drawn. 

The  shears  themselves  are  readily  found  from  the  moment  lines.    The  left  reaction  is 

equal  to  the  ordinate  to  the  equilibrium  polygon  at  the  right  abutment,  divided  hy  the  length 

of  the  truss.    The  panel  load  to  be  subtracted  is  equal  to  the  similar  ordinate  at  the  right  end 

of  the  panel,  divided  by  a  panel  length.*  These  ordinates  are  measured  from  the  line  o-O  if 

there  are  no  wheels  to  the  left  of  the  left  abutment  or  the  left  panel  point.    If  there  are,  then 

the  ordinates  are  measured  from  the  moment  line  corresponding  to  the  wheel  nearest  the 

bridge  or  panel  on  the  left. 

Example  i.  Find  the  maximum  stresses  in  all  the 
members  of  the  truss  of  Art.  96. 

Example  2.  Find  the  maximum  moments  and  shears 
at  points  10  ft.  apart  in  a  plate  girder  100  ft.  long. 

lOi.  Computation  of  Panel  Concentrations. 

— It  is  frequently  required  to  compute  all  the 
panel  loads  in  a  truss  for  some  single  position  of 
the  loading;  as,  for  example,  with  wheel  3  at  the 
first  panel  point  from  the  left  end.  A  convenient 
formula  is  readily  derived  as  follows  : 
„_,_,,  etc.,  be  the  panel  concentrations ;  ,  M^,  .  .  . 
^,  ,  etc.,  the  summation  of  the  moments  of  the  loads  to  the  left  of  each  panel  point  about 

*  The  student  must  remember  that  in  truss  analysis,  for  both  shears  and  moments,  the  wheel  loads  must  first  be 
concentrated  at  the  joints  before  they  can  come  upon  the  truss,  and  when  the  method  of  sections  is  employed  ;'oint 
loads  only  are  to  be  considered.    The  external  reactions,  however,  can  be  computed  from  the  actual  positions  of  the  loads. 


2 


n-1 


P/i+i 


Fig.  135. 


In  Fig.  135,  let /'„,/', 


P  P 


MODERN  FRAMRD  STRUCTURES. 


that  point;  b,  the  distance  from  panel  point  n  to  any  other  panel  point;  and  d,  the  panel 
length.  Suppose  P„  is  required.  By  equating  the  moments  of  the  loads  about  n  with 
the  moments  of  the  panel  reactions  we  have 

:2r'Pb  =  M,  ....(«) 

Similarly  with  centre  at  «  -f-  i, 

:s:'P{b  ^d\  =  :2:-  ^pt + d^:  -^p+pa  =  m„+,,  {t) 

and  with  centre  at  «  —  i, 

2:  -  ^Pib  -  d),  =  2:' -^Pb-d2:-^p,=M„.,  ic) 

Substituting  from  {a)  and  {c)  in  {b)  we  readily  get 


P.  = 


(8) 


The  values  of  the  J/'s  can  be  taken  from  the  diagram,  or  may  be  computed  with  the  aid  of 
the  moment  table,  Fig.  129.    For  an  example  of  the  application  of  eq.  (8),  see  Art.  107. 


TRUSSES  WITH  INCLINED  WEB  MEMBERS  AND  INCLINED  CHORD.S. 

102.  Maximum  Moment  at  Joints  in  the  Unloaded  Chord.— Let  AB,  Fig.  136,  be 
any  truss  with  horizontal  or  inclined  chords,  and  C  any  joint  of  the  unloaded  chord,  not  verti- 




 yrr  

1 

E 

F 

2/1 

H 

^.<— 

G 

 y 

1  1 

1 
1 
1 
1 

D' 


E' 

Fig.  136. 


cally  over  a  joint  of  the  loaded  chord.  Let  b  =  the  horizontal  di.stance  from  C  to  the  next 
loaded  chord  joint  towards  A,  and  a  —  the  horizontal  distance  from  C  to  A.  The  other 
notation  is  the  same  as  that  previously  used.  The  effect  upon  the  moment  at  C  due  to  a  load 
unity  moving  from  B  to  E  and  from  D  to  A  is  the  same  as  for  the  moment  at  a  point  (7  in  a 
beam  ;  hence  the  influence  line  for  that  portion  is  B'G  and  FA',  portions  of  the  influence  line 

AHB',  where  C'H  —  — — Between  E  and  D  the  load  is  carried  by  the  stringer  to  these 

panel  points.    The  influence  line  for  this  portion  will  then  be  the  straight  line  GF. 

To  derive  the  criterion  for  maximum  moment  at  C,  let  G,,  G^,  and  G,  represent  the 
portions  of  the  load  that  lie  in  the  segments  AD,  DE,  and  EE,  respectively.  Let  the  loads 
advance  a  small  distance  Sx  to  the  left.    The  increase  in  moment  at  C  will  be 


For  a  maximum, 


SM  =  G^Sx  tan  ^3  —  G^Sx  tan  —  G^Sx  tan  or, . 
dM 


Sx 


=  0—6^,  tan      —  G^  tan  «,  —  G^  tan  or, 


BRIDGE-TRUSSES    ANALYSIS  FOR  WHEEL-LOADS.  89 

Now  we  have 

a                I -a                GE'  -  FD 
tan  -j\    tan  «,  =  — j— ;    tan     =  ^  . 

Also 

GE'  =  [/  —  {a  -  b     d)]  tan  and    ED'  =  [a  -  b)  tan  «, . 

Substituting  these  values  of  GE'  and  ED'  in  the  expression  for  tan  a,  we  have 

bl  -  ad 

tan  a,  = 

Eq.  («)  then  becomes 

a         bl—ad      ^^  —  a 


If     =  G^,  +  G^,  +  G^3 ,  we  have,  after  reduction, 

i   ^  =o»  (*) 


or 

^   (9) 


For  a  maximum  the  left-hand  member  of  eq.  {p)  must  become  zero  by  passing  from 
positive  to  negative.    This  can  occur  only  when  some  wheel  passes  E  or  D,  as  then  only  can 


be  increased.    Equation  (9)  is  the  criterion  required. 


Fig.  137. 


I^ig-  ^  37  shows  a  load  line  and  the  outline  of  a  truss  AB.  Try  wheel  4  at  Z)  for  a  maximum 
moment  at  C.    The  slope  of  the  line  A' T  is  equal  to  the  left-hand  member  of  eq.  (9). 


90 


MODERN  FRAMED  STRUCTURES. 


varies  between  the  values  GM  and  GM' ,  or  CK  and  CK' ,  according  as  wheel  4  is  considered 
in  or  out  of  the  panel  DE.    The  value  of  (7,  varies  correspondingly  between  the  sum  of  the 

loads  6,  5,  and  4,  and  the  sum  of  6  and  5  onl)-.    The  corresponding  values  of  G^^  are  found 

by  drawing  MO  and  M'O.  They  are  ATiV  and  K N' .  The  slopes  of  the  lines  yiW  and /^W,  if 
drawn,  would  therefore  be  the  two  values  of  the  second  member  of  eq.  (9).  Hence  when 
-'J'T'cuts  AW,  then  this  position  of  the  loads  gives  a  maximum  moment  at  C.  None  of  the 
lines  need  be  drawn,  the  points  N  and  N'  being  lightly  marked  and  a  thread,  A'T,  stretched 
from  A'  to  the  intersection  of  the  vertical  at  B  with  the  load  line.  With  a  load  at  E,  two 
points  similar  to  iV  and  N'  are  obtained  by  lines  drawn  from  the  intersection  of  the  vertical 
through  D  with  the  load  line,  to  the  upper  and  lower  limits  of  the  load  at  E. 

The  position  of  the  loads  having  been  found,  the  bending  moment  at  C  is  most  readily 
found  by  proportion  from  the  moments  at  D  and  E.  Thus  if  and  are  the  bending 
moments  at  D  and  E,  respectively,  found  in  the  usual  way,  then  the  moment  Mc,  at  C, 

103.  Maximum  Web  Stresses. — Let  ED,  Fig.  138,  be  any  web  number  of  a  truss  with 
inclined  chords.    If  /  is  the  intersection  of  the  chord  members  EF  a.nA  CD,  then  the  stress 


Fig.  138 

in  ED  is  equal  to  the  sum  of  the  moments  about  /  of  all  the  external  forces  acting  upon 
the  portion  to  the  left  of  pq,  divided  by  the  lever-arm  t.  The  stress  in  ED  is  then  a  maxi- 
mum when  this  moment  is  a  maximum. 

The  influence  line  for  moment  at  /  is  readily  drawn,  after  first  finding  the  values  of  this 

moment  for  a  load  unity  at  D  and  at  C.    With  load  unity  at  D,  the  moment  at  /  =  - — —. — -s. 


BRIDGE-TRUSSES— ANALYSIS  FOR  WHEEL-LOADS, 


and  with  the  load  at  C  the  moment  =  -"y-^  —     +  A  *  negative  quantity.    Laying  off  D'G 

and  C'H  equal  respectively  to  these  moments,  and  joining  A' H,  HQ,  and  GB' ,  we  have  the 
required  influence  line  for  moment  at  /. 

The  maximum  stress  in  ED  occurs  when  some  of  the  wheels  at  the  head  of  the  train  are 
in  the  panel  CD^  and  in  exceptional  cases  only,  when  some  of  the  loads  are  to  the  left  of  C. 
In  the  cases  treated  we  will  assume  that  there  are  no  loads  on  the  portion  AC. 

Let  represent  the  total  load  in  the  panel  CD;  G^,  the  load  in  DB\  and  G,  the  total 
load  on  the  bridge.  Let  the  loads  advance  from  any  given  position,  a  distance  6x  towards 
the  left.    The  moment  at  /  will  be  increased  by  an  amount 

6M  =  G^6x  tan  a,  —  G^^Sx  tan  a, ; 

and  for  a  maximum, 

6M  . 

=z  o  =  G^  tan  a,  —  G^  tan  a,  (a) 

D'G  s 

Now  tan  a,  =  _^=^,  ...(*) 

and 

ds 

D'G -C'H     -7  +  ^  +  -^ 
tana.  =  ^         =  ^   {c) 

Substituting  from  {b)  and  {c)  in  («),  we  have,  by  putting  G  for  G.^-\-  G^, 
or 



as  the  criterion  for  maximum  moment  at  /  or  maximum  stress  in  ED. 

This  criterion  is  nearly  the  same  as  that  for  maximum  shear  in  Art.  94  :  differing  only  in 

a 

that  here  the  load  in  the  panel  is  to  be  increased  by  the  -th  part  of  itself  before  dividing  by 

the  panel  length.  This  correction  should  be  added  to  the  ordinate  for  G^  when  using  the 
diagram,  before  the  thread  is  stretched. 

For  the  member  EC  the  same  panel  CD  is  partially  loaded  and  s  is  replaced  by  /,  while 
the  other  quantities  remain  the  same  as  for  ED. 

The  actual  stress  in  ED  is  most  readily  obtained  by  first  determining  the  horizontal 
components  in  .£/^and  CD,  as  was  done  in  Chap.  IV,  Art.  80. 

For  EC  or  for  any  web  member  where  verticals  are  not  used,  it  is  easier  to  get  the 
moments,  about  /,  of  the  abutment  reaction  at  A  and  the  panel  load  at  C,  and  then  divide  by 
the  lever-arm  of  the  member.  This  reaction  and  panel  load  are  obtained  from  the  summa- 
tion of  moments,  or  from  the  diagram,  as  before. 

104.  Application  of  the  Foregoing  Principles. — The  truss  of  Fig.  109.  p.  68.  liaving 
both  inclined  web  members  and  inclined  chords,  will  serve  to  illustrate  the  principles  of  the 
two  preceding  articles.  The  stresses  in  all  the  members  will  be  found,  using  the  loads  of  the 
diagram  Fig.  131. 

1st.  Upper  Chords. — The  truss  should  first  be  drawn  to  a  large  scale,  so  that  any  required 
lever-arm  can  be  scaled  off  with  sufficient  accuracy. 


92 


MODERN  FRAMED  STRUCTURES. 


The  centres  of  moments  for  the  upper-chord  members  are  at  the  lower  chord  joints. 
These  moments  are  found  as  in  a  beam  or  a  Pratt  truss,  Art.  96,  and  hence  the  computations 
will  not  be  given  here  in  detail.  The  table  of  chord  stresses  below  gives  the  positions  of 
loads  for  the  greatest  maximum  moments,  the  corresponding  moments,  the  lever-arms  of  the 
various  chord  members,  and  the  resulting  stresses. 

When  the  piece  MN  is  reached,  the  centre  of  moments  is  at  E  or  F  according  as  EN  or 


yf"^     /  \  ^ 

/\    ^      vs.  ®' 

/  '?"\  /   \  / 

/       '""X  /          \  / 

7^.  / 

'  A? 
/  \ 

/  A 

^**^  /    \      »o  /!\  ^^"^ 

1  \    53/  1  \  _t  /|\ 

/ ^\  f\  \  \y \  \ 
Y  i  1  yi  i  Y  j  \ 

I A -135.5         C-192.0  D- 

1 
1 

U  

-2IS.O  E 

-226.0 
 -200.0 

L.^jG         'l«-13;8'4"  i*-20;0^ 
8.25' 

Fig.  139. 


MF  is  in  action.  Trying  the  point  E,  we  have  as  our  maximum  moment  8950.  The  cor- 
responding shear  in  the  panel  EF  is  found  by  subtracting  from  the  reaction  at  A  the  loads 
between  A  and  E  and  the  portion  of  the  loads  in  EF  going  to  E.  It  is  found  to  be  —  12.3  ; 
hence  EN  must  be  in  action  and  the  proper  centre  of  moments  for  this  position  of  the  loads 
is  as  we  have  assumed.  The  maximum  moment  at  F  is  8760,  which,  being  less  than  8950,  is 
not  considered.    Hence  the  maximum  stress  in  MN  is  due  to  a  moment  at  E  of  8950. 

2d.  Lower  CJiords. — The  centres  of  moments  for  the  lower  chord  members  are  at  the 
upper  chord  joints.  The  positions  of  the  loads  giving  maximum  moments  at  these  joints  are 
found  by  means  of  the  diagram  Fig.  131,  as  explained  in  Art.  102. 

Moment  at  /.—This  is  a  maximum  for  wheel  4  at  C.  The  moment  is  found  by  propor- 
tion, from  the  moments  at  A  and  C  {M^  and  Mc).  M,.,  =  o.  Mc  is  readily  found  from  the 
diagram  Fig.  131,  by  reading  off  the  ordinate  to  the  curved  moment  line  at  the  right  end  of 
the  truss,  dividing  by  200,  multiplying  by  28.57,  and  subtracting  the  ordinafe  to  the  moment 
curve  at  C.  Thus, 

35000 


Mr 


200 


X  28.57  —  360 


35000 

7 


X  I  —  360  =  4640. 


20 


Then  J//  =  M^-\-{Mc  -  ^a)^^  =  3250  as  given  in  the  table.    The  lever-arm  oi  AC=2^  ft. 

Moment  at  K. — Wheel  9  at  Z>  and  wheel  4  at  C  are  the  two  positions  giving  maximum 
moments. 


With  wheel  9  at 


and 


With  wheel  4  at  C 


=  34500  X  f  —  2400  =  7460, 

Mc  =  34500  X  4-  -    300  =  4630, 

J  5  g 

J/a'  =  4630  +  (7460  -  4630)^  -  6000. 


=  35000  X  f  —  2600  =  7400, 

Mc        35000  X  I  —     360  =  4640, 


BRIDGE-TRUSSES— ANALYSIS  FOR  WHEEL-LOADS. 


93 


and 

J  5  g 

=  4640  +  (7400  -  4640)^  =  5970- 

The  maximum  moment  is  therefore  6000,  and  this  divided  by  the  lever-arm  31.35,  gives  the 
stress  in  CD,  which  equals  192  thousand  lbs. 

Moment  at  L. — Wheel  8  at  and  wheel  13  at  are  the  two  positions  giving  maxima 
The  greatest  of  the  two  moments  is  when  wheel  8  is  at      and  is  equal  to  7900. 

Moment  at  M  or  N  for  Stress  in  EF. — When,  for  any  position  of  the  loads,  the  moment 
at  E  is  greater  than  the  moment  at  F,  the  shear  in  panel  EE  is  negative,  and  hence  EN  is 
in  action  ;  and  when  the  moment  at  /'"is  greater  than  the  moment  at  E,  then  MF  is  in  action. 
This  follows  readily  from  shear  =  differential  of  the  moment,  or  from  the  segment  under  EF 
of  the  equilibrium  polygon. 

*  Wheels  12  and  13  at  i;'  are  found  to  give  maximum  moments  at  M,  but  in  each  case  the 
moment  at  E  is  greater  than  the  moment  at  E,  and  therefore  the  centre  of  moments  for  EF 
is  not  at  M. 

Wheels  14,  15,  and  16  at  F  are  found  to  give  maximum  moments  at  N,  and  moreover 
with  15  at  the  moments  at  E  and  each  equal  8760.  This  is  evidently  the  greatest 
possible  moment  at  F  that  is  not  greater  than  the  corresponding  moment  at  E,  and  hence 
the  maximum  moment  at  N  is  also  8760. 

The  following  table  gives  the  chord  stresses  in  thousands  of  pounds. 


LIVE  LOAD  CHORD  STRESSES. 


Member. 

Centre  of 
Moments. 

Position  of 
Loads. 

Maximum 
Moment. 

Lever-arm. 

Stress. 

IK 

c 

4  at  C 

4640 

25-5 

+  182 

KL 

D 

8  at  Z» 

7470 

33-7 

-j-  221 

LM 

E 

13  at  E 

8950 

38.1 

+  235 

MN 

E 

13  at  E 

8950 

38.7 

+  231 

AC 

I 

4  at  C 

3250 

24.0 

-  135-5 

CD 

K 

g  at  Z) 

6000 

31-3 

—  192 

DE 

L 

8  at  /? 

7900 

36-3 

—  218 

EF 

N 

15  at  E 

8760 

38.7 

—  226 

3d.  JVelf  Stresses. — We  first  produce  the  upper  chord  members  to  their  intersection  with 
the  lower  chord,  and  scale  off  the  distances  of  these  intersections  from  the  point  A.  These 

are  the  quantities  " s"  of  Art.  80.    The  ratios  —  are  given  in  the  third  column  of  the 

following  table  of  web  stresses.    They  need  be  computed  only  to  the  nearest  tenth. 

The  horizontal  component  of  the  maximum  stress  in  AI  is  equal  to  the  maximum  stress 

in  AC  —  I35-5.    For  IC,  the  ratio  —  =  o,  and  the  same  position  of  loads  gives  a  maximum 

stress  in  this  member  as  in  AC\  that  is,  wheel  4  is  at  C.  The  horizontal  component  of  stress 
in  IC  —  hor.  comp.  IK  —  hor.  comp.  AC.  To  find  these  horizontal  components  we  need  the 
moments  at  A,  /,  and  C.  The  moments  at  A  and  C  are  given  in  the  fifth  and  sixth  columns  of 
the  table  below.  The  inoment  at  /,  found  by  proportion  from  these  two  moments,  is  given 
in  the  next  column.  The  hoi  comp.  in  IK,  found  by  dividing  the  moment  at  C  by  the 
vertical  distance  from  C  to  IK,  s  given  in  column  8.  The  stress  in  AC  is  given  in  the  next 
column,  and  the  difference  between  columns  9  and  8  is  the  hor.  comp.  in  IC.  The  actual 
stress  is  given  in  the  last  column. 

a 

For  the  member  CK,  -  =  0.6.    In  testing  for  the  position  of  the  loads  we  then  multiply 


94 


MODERN  FRAMED  STRUCTURES. 


the  loads  in  the  panel  CD  by  1.6,  and  use  this  product  in  the  place  of  the  loads  themselves. 
(See  Art.  80.)  Wheels  2  and  3  at  Z>  are  two  positions  giving  maxima.  Only  the  greater  one 
is  given  in  the  table.  The  stress  in  CK  is  found  in  the  same  way  as  was  that  in  IC,  that  is,  by 
finding  the  horizontal  components  of  the  stresses  in  IK  and  CD.  The  stresses  in  KD,  LE, 
MF,  and  NG  are  found  in  like  manner. 

The  members  LD  and  ME  being  so  nearly  vertical,  it  will  be  more  accurate  to  find  the 
stresses  in  these  pieces  by  finding  their  vertical  components. 

The  vertical  component  in  LD  =  shear  in  DE  —  vert.  comp.  KL,  =  left  reaction  —  load 
at  D  —  vert.  comp.  KL.  The  vertical  component  in  KL  is  readily  found  from  its  horizontal 
component,  and  the  other  quantities  are  easily  computed.  Likewise  the  vert.  comp. 
ME  =  shear  in  EF  —  vert.  comp.  LM. 

The  horizontal  components  of  the  stresses  in  KL  and  LM  are  given  in  column  8  of  the 
table. 

The  stresses  are  given  also  along  the  members  in  Fig.  139. 

LIVE  LOAD  WEB  STRESSES. 


Moments  at  Lower 

Member. 

Length. 

a 

Position  of 

Chord. 

Moment  at 

Hor.  Comp. 

Hor.  Comp. 

Hor.  Comp. 

Stress  in 

s 

Loads. 

Up.  Chord. 

Up.  Chord. 

Lower  Ch'd. 

Web. 

Web. 

Left. 

Right. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

10 

II 

AI 

31.2 

4  at  C 

135-5 

135-5 

+  212. 

IC 

25-5 

0 

4  at  C 

0 

4640 

3250 

173.1 

135-5 

37-6 

-  III. 5 

CK 

34-3 

.6 

3  at 

3520 

6840 

5122 

131-3 

163. 1 

31.8 

+  79-0 

KD 

34.8 

•3 

3  at  Z) 

3520 

6840 

5122 

197-7 

163. 1 

34-6 

—  81.4 

LE 

41.6 

.2 

3  at  j5 

4846 

7069 

5487 

184. 1 

151 .2 

32-9 

-  67-4 

MF 

46.3 

0 

4527 

5836 

4669 

150.8 

120.7 

30. 1 

-  54-7 

NG 

50.0 

—  .2 

2  at  G 

2662 

3268 

2598 

92. 1 

67.1 

25.0 

-  39-5 

Vt.  Comp. 

LD 

37-2 

.6 

2  at  £ 

4460 

128.9 

48-4 

4-  49-6 

ME 

38.8 

•3 

2  at  A 

4050 

105.4 

34-3 

+  34-5 

105.  The  Petit  Truss. — The  maximum  stress  in  the  piece  EF,  Fig.  140,  is  equal  to  the 
maximum  moment  at  D,  divided  by  its  lever-arm.    The  maximum  stresses  in  the  members 


Fig.  140 


HG  and  LLC  are  due  to  the  maximum  joint  load.  The  piece  ND  when  stressed  is  the  only 
web  member  in  action  in  the  panel  GD,  and  hence  its  maximum  stress  is  determined  accord- 
ing to  Art.  103.  The  same  applies  to  the  counter,  NF.  There  remain  then  to  be  treated 
only  the  pieces  CD,  EC,  EH,  and  CH  when  acting  with  the  counter  HF. 


BRIDGE-TRUSSES— ANALYSIS  FOR  WHEEL-LOADS. 


95 


1st.  The  piece  CD. — The  centre  of  moments  for  CD  is  at  E,  but  the  joint  G  is  on  the 
left  of  the  section.    The  influence  line  for  the  portions  AC  and  DB  will  be  A' K  and  MB' , 
ail  —  a) 

where  KC  =  ^  ,  which  is  the  same  as  the  ordinary  influence  line  for  moment  in  a  beam  ; 

but  as  the  unit  load  moves  from  C  to  G,  the  moment  about  E  of  the  forces  on  the  /eft  of  the 
section  is  uniformly  increased  at  the  same  rate  as  from  A  to  C.  Hence  KL  will  be  a  prolon- 
gation of  A'K.    From  G  to  D  the  influence  line  is  the  straight  line  LM. 

To  determine  the  position  for  a  maximum  moment  at  E :  Let  G, ,  G,,  G^  ,  and  G  be 
respectively  the  load  on  AG,  GD,  DB,  and  the  entire  bridge. 

A  small  movement  of  the  loads  to  the  left  will  increase  the  moment  by  an  amount 
equal  to 

dM  =  G^dx  tan  a,  -\-  G,Sx  tan  a,  —  G^Sx  tan  a, . 

For  a  maximum, 

^                ^  ^ 
O  =:      tan  o-,  -|-  G^  tan      —  u,  tan  or,  {a) 


Now  we  have 


and 


6x 


a  I  —  a 

tan  «,  =  y ;    tan  or,  =  — — ; 


MN    d  tan  a,  +  2d  tan  a.     I  —  a     2a  a 

Substituting  these  values  in  {a)  and  reducing,  we  have  as  the  condition  for  a  maximum, 

G  G,-G„ 


I  a 


00 


To  satisfy  this  criterion  a  load  must  be  at  G. 

Eq.  (11)  is  easy  of  application  by  means  of  either  the  table  or  diagram. 

The  moment  at  E  is  equal  to  the  reaction  at  A  multiplied  by  AC,  minus  the  moment 
of  the  wheel-loads  to  the  left  of  C  about  C,  plus  the  moment  of  the  joint  load  at  G  about  C. 
This  joint  load  is,  by  Art.  91,  equal  to  the  bending  moment  in  the  centre  of  a  beam  of  length 
2 

2d,  multipUed  by  ^.    Hence  the  moment  of  the  joint  load  G  about  C  is  equal  to  twice  the 

bending  moment  at  G  if  CD  were  a  stringer.  In  other  words,  the  total  moment  at  E  is  equal 
to  the  bending  moment  at  67  in  a  beam  AB,  plus  twice  the  bending  moment  at  G  in  the 
beam  CD. 

2d.  The  piece  EH. — When  the  stress  in  this  piece  is  a  maximum,  HF  is  not  in  action. 
Now  for  any  position  of  the  loads,  the  stress  in  EH  would  not  be  altered  if  HD  and  ED  were 
to  be  made  separate  parallel  pieces,  as  the  portion  CHD  would  then  form  a  separate  small 
truss.  But  this  small  truss  serves  merely  the  ofifice  of  a  floor-stringer  in  carrying  loads  to  the 
joints  6  and  D.  Hence  the  stress  in  EH  is  the  same,  and  the  position  of  loads  for  a  maxi- 
mum stress  is  the  same,  as  for  ED  were  the  pieces  CH  and  HG  removed.  This  case  reduces 
then  to  the  case  of  Art.  103,  in  which  the  panel  length  is  CD. 

3d.  The  piece  EC. — For  this  piece,  as  for  EH,  we  may  consider  CH  and  HG  removed 
and  proceed  as  in  Art.  103. 

4th.  The  piece  CH  when  acting  as  a  tie. — Here  the  member  HD  is  not  in  action.  The 
same  process  applies  to  CH  as  applied  to  EH.  That  is,  consider  EH  and  HG  removed,  and 
find  the  maximum  stress  in  CF.  This  will  be  the  maximum  in  CH,  but  for  the  maximum  in 
HF  the  panel  to  be  considered  is  GD. 


96 


MODERN  FRAMED  STRUCTURES. 


DOUBLE  SYSTEMS. 

I06.  The  influence  line  for  either  chord  or  web  stress  in  a  truss  with  a  double  system  of 
bracing  is  made  up  of  straight  Hues  of  many  different  inclinations.  Since  the  criteria  for 
maximum  values  contain  as  many  terms  as  there  are  different  inclinations  in  the  influence 
lines,  in  this  case  they  are  very  difficult  of  application. 

For  example,  take  the  Whipple  truss  of  Fig.  141.  The  centre  of  moments  for  the  chord 
member  FH  for  loads  on  the  full  system  is  at  C.  For  loads  at  the  joints  of  this  system  the 
ordinates  for  moment  at  C  are  ordinates  to  the  influence  line  A' IB' .  While  for  loads  on  the 
dotted  system,  the  centre  of  moments  for  FH  is  at  D,  and  the  ordinates  for  moment 
at  D  are  ordinates  to  A' KB' .  Hence-  the  influence  line  for  stress  in  FH  is  the  broken  line 
A' abcIKde. .  . .  B'.  The  points  a  and  z  are  taken  half-way  between  A' KB'  and  A' IB',  assuming 
the  loads  at  J/ and  A^to  be  equally  divided  between  the  two  systems. 

The  influence  line  for  shear  in  panel  CE  of  the  full  system  is  the  full  broken  line  of  the 
lower  figure,  the  ordinates  to  this  line  being  zero  for  the  loads  at  joints  of  the  dotted  system. 
Loads  at  J/ and  A' are  equally  divided. 

The  influence  line  for  shear  exhibits  in  a  striking  manner  the  effect  of  concentrated  loads 
upon  the  web  members  when  the  distance  between  the  loads  is  an  even  number  of  panel 
lengths.  As,  for  example,  two  engine  concentrations  when  the  panel  lengths  are  twenty-five 
feet,  the  length  of  a  locomotive  being  about  fifty  feet.    Panels  of  such  length  are  therefore. 


Fig.  141. 

by  all  means,  to  be  avoided  in  a  double  system,  and  a  length  chosen  which  will  bring  the  con- 
centrations upon  different  systems. 

The  cumulative  effect  upon  chord  members  is  seen  to  be  very  small. 

In  computing  stresses  in  a  double-intersection  tru.ss,  either  an  equivalent  uniform  load 
should  be  taken,  according  to  the  method  of  Chapter  VI,  or,  as  is  often  done,  such  a  position 
of  the  loads  should  be  selected  as  will  give  the  maximum  joint  load  at  the  end  panel,  and  this 
same  position  retained  for  all  the  chord  stresses.  F'or  web  stresses  the  load  should  be  moved 
one  panel  to  the  right  each  time,  and  the  portion  of  the  load  going  to  the  joint  in  front 
neglected.  The  stresses  are  then  obtained  by  computing  the  panel  concentrations  for  these 
positions  by  Art.  lOi  and  treating  each  system  separately. 


BRIDGE-TRUSSES— ANALYSIS  FOR  WHEEL-LOADS. 


97 


By  placing  the  loads  as  above  indicated  only  one  set  of  concentrations  is  required. 
This  method  is  applicable  to  either  parallel  or  inclined  chords.  In  the  latter  the  chords  are 
to  be  considered  as  straight  between  successive  joints  of  the  system  under  consideration. 

A  more  accurate  method  for  chord  stresses  would  be  to  find  the  position  of  the  loading  as 
for  moment  in  a  single-intersection  truss,  then  compute  the  corresponding  panel  concentra- 
tions and  find  the  stress  in  the  member  in  question,  as  above  explained.  This  would  involve 
the  computation  of  panel  concentrations  for  each  different  position,  but  would  probably  give 
stresses  very  near  the  exact  maximum. 

107.  Stresses  in  a  Whipple  Truss. — Let  us  take,  for  example,  the  truss  of  Fig.  142. 
Span  =:  272  ft.  ;  panel  length  =  d  =  17  ft.  ;  height  =  ^  =  38  ft.  Loading  to  be  as  given 
in  the  diagram  and  table.  The  method  of  panel  concentrations  will  be  used,  one  position  of 
the  loads  being  taken  for  all  chord  stresses. 


BCDEFG  HIJKLIVINOP 


a      bcdefghij       k      I      m      n      o     jp  q 


Fig.  142. 


1st.  Panel  Concentrations. — Wheel  4  at  3  is  found  by  trial  to  give  the  maximum  load  at  i>. 

M„     —  2M„  +  M, 

By  eq.  (8),  p.  88,  any  panel  concentration,  P„,  —  — '—  "^-^  '— ,  where  M„,  M„^,, 

and  M„  _ ,  are  the  summations  of  moments  at  the  joint  in  question  and  the  ones  to  the  right 
and  left  respectively,  and  c/  is  the  panel  length. 

With  wheel  4  at  d  we  have  the  following  positions : 


Wheel.  Joint. 

4  ^ 

7  ^ 

9  d 


Distance  of 
Wheel  to  the 
left. 

0.0 

0.6 

7.0 


Wheel. 

12 

15 


Joint. 


.e 

f 


end  of  uniform 
load 


Distance  of 
Wheel  to  the 
left. 

1.9 

2.8 

0.4 


Calling  joint  a  zero,  we  have  the  following  values  of  J/ with  wheel  4  at  ^: 


=  0; 

M, 

=  369.2  —  8  X  18.4  =  222.0; 

=  1460. 1  +   86  X  0.6  —  8  X  35.4 

=  1228.5 ; 

M, 

=  2414.9  -f-  104  X  7-0  —  8  X  524 

=  2723.7; 

=  491 1.5  +  142  X  1-9  —  8  X  69.4 

=  4626. 1  ; 

=:  7478.2  +  181  X  2.8  -  8  X  86.4 

=  7293-8 ; 

=  1 1236.2  +  208  X  0.4  +  X 

1.5  -  8  X  103.4  = 

10492.3 

(17)' 

=  10492.3  +  (200  +  0.4  X  1.5)  X  17  +    2    X  '-5  - 

14119-3 

9&  MODERN  FRAMED  STRUCTURES. 

The  Values  of  P;  beginning  at  /*„ ,  are : 


222.0  ^ 

P.  =  -jy-  =  13.06; 


„      1228.5  —  2  X  222.0 

=  =  46.15 ; 

2723.7  -  2  X  1228.5  +  222.0 

=  —   =  28.75 ; 


=  :[j   =  23.95  ; 

7293-8  -  2  X  4626.1  +  2723.7 

p  =   =  45-02; 

_  10492.3  -  2  X  7293-8+4626.1  _  „. 

^6  —  3''22, 

p  _  141 19-3  -  2  X  10492-3  4-  7293-8  _ 

•    .m    25.21. 


Beyond  the  concentrations  are  each  equal  to  a  panel  length  of  uniform  load  =: 
17  X  1.5  =  25.5. 

2d.  Wel>  Stresses. — The  secant  of  the  angle  which  the  members  aB  and  Be  make  with  the 
vertical  =     1'^"  -\-  38''  ^  38  =  1.063;  and  that  of  the  angle  whicli  the  remaining  diagonals 

make  with  the  vertical  =  I  34'  +  38'  38  =  1.234.  The  loads  at  b  and  p  will  be  assumed  to 
be  equally  divided  between  the  two  systems. 

The  stress  in  nB  is  found  from  the  shear  in  ai>,  with      at  d.    We  have  then 

stress  in  aB  =  1.063  x  tVC^S  X  46.15  +  H  X  28.75  +  '3  X  23.95  +  12  X  45-02 

+  II  X  31-22+  10  X  25.21  +  (9  +  8+  ...  +1)  X  25.5]  =  245.1. 

For  Be,  /*,  is  placed  at  c ;  whence 

stress  in  Be  =  1.063  x  rVlM     46- 15  f  12  X  23.95  -|-  10  X  31.22 

+  (8  +  6  +  4+2  +  i)X  25.5]=  1 17.5. 

For  cC  and  Ce,      is  at  e,  and  we  have 

stress  in  cC  —  ^-^{\2  X  46.15  +  10  X  23.95  +  8  X  31.22  +(6  +  4  +  2  +  ^)  x  25.5]  =  85.2- 
stress  in  Ce  —  1.234  X  85.2  =  105.1. 

For  Bd,  P,  is  at  d,  and 

stress  in  Bd  -  1.234  X  yVC'S  X  46.15  +  n  X  23.95  +  9  X  31.22 

+  (7  +  5+3+i)  X  25.5]  =  118.7, 

For  dD  and  /;/.  P,  is  at  /;  etc. 

The  maximum  stress  in  Bb—  P,  —  46.1  5. 

3d.  Chord  Stresses. — For  all  chord  stresses  P^  is  at  b. 

245.1  17 

The  stress  in  ac  —  hor.  comp.  aB  —  — X  „  =  10^.2. 

^  1.063  38 


BRIDGE-TRUSSES— ANALYSIS  FOR  WHEEL-LOADS. 


99 


The  stress  in  any  other  chord  member,  as  BE,  is  found  by  obtaining  separately  the 
portion  due  to  each  system  and  then  adding  the  two  results.  For  the  full  system,  the  centre 
of  moments  is  at  e,  and  for  the  dotted  system  it  is  at  /. 

Abutment  reaction,  full  system 

=  TV[i  X  46.15  X  15  +  14  X  28.75  +  12  X  45-02+IO  X  25.21 

+  (8  +  6+4  +  2  +  i)  X  25.5]  =  129.0; 

whence,  stress  in  DE  due  to  loads  on  the  full  system 

=  [129.0  X  68  -(i  X  46.15  X  51  +28.75  X  34)]  ^  38  =  174-2. 
Abutment  reaction,  dotted  system, 

=  ^[i  X  46.15  X  15  +  13  X  23.95  4-  II  X  31-22  +(9+7  +  5  +  3  -fi)  X  25.5]  =  101.6, 
and  hence  the  stress  in  DE  due  to  loads  on  the  dotted  system 

=  [  101.6  X  85  -  (i  X  46.15  X  68  +  23.95  X  34)]  ^  38  =  i64.5- 
The  total  stress  in  DE  or  7^  therefore  =  174.2  -f  164.5  ~  338- 7- 

The  stresses  in  the  other  chord  members  are  found  in  a  similar  manner,  using  the 
reactions  above  found.  Near  the  centre  of  the  truss  it  may  be  necessary,  in  selecting  the 
centre  of  moments,  to  find  the  sign  of  the  shear  in  one  or  both  of  the  systems  in  order  to 
know  which  diagonal  is  in  action.  This  is  easily  done,  as  the  abutment  reaction  and  panel 
loads  are  already  known. 

SKEW-BRIDGES. 

108.  Fig.  143  illustrates  a  skew-bridge  similar  to  that  of  Figs.  1 19-121,  p.  71.  As  in  Art.  86, 
the  loads  may  be  considered  as  applied  along  the  centre  line  XY.  As  an  example,  let  us  find 
the  influence  line  for  moment  at  F.  Loads  from  c  to  /  will  evidently  have  the  same  effect 
upon  the  truss  AB  as  if  the  bridge  were  square  and  equal  to  AB  in  length.  The  influence 
line  from  c  to  i  will  then  be  the  portion  c"f"i",  Fig.  144,  of  the  influence  line  A"f"  B"  for 
moment  at /in  a  beam  of  length  AB.  Beyond  c"  and  i"  the  line  is  a  straight  line  with  zero 
ordinates  at  a'  and  b' . 

From  this  influence  line  the  criterion  for  maximum  moment  may  be  written  out.  It  is 
seen,  however,  that  the  true  influence  line  differs  but  slightly  from  A"f"B'\  and  as  regards 
the  position  of  the  loads,  it  may  be  assumed  the  same.  Hence  for  a  maximum  moment  at  F 
the  average  unit  load  on  length  AF  rnw^l  be  equal  to  that  on  length  FB. 

For  moment  at  the  point  /  the  influence  line  would  be  straight  from  i  to  c,  and  in  that 
case  it  would  be  more  exact  to  make  the  average  load  on  the  length  ^/ equal  to  that  on  ib. 

The  influence  line  for  moment  at  /,  Fig.  143,  for  finding  the  stress  in  CD,  is  a  C" D" i"b\ 
Fig.  145  ;  and  is  drawn  by  drawing  the  influence  line  A  " I" B"  for  moment  at  /  in  a  beam  of 
length  AB,  and  then  drawing  the  straight  lines  a  C" .  C  ' D"  and  i"b' . 

For  moment  at  /  for  stress  in  AC  influence  line  is  a'l"'i"b' .  In  any  case  the  influ- 
ence line  may  be  drawn  and  the  exact  criterion  worked  out  if  desired  ;  or  an  approximate  one 
found  by  drawing  such  a  line  of  the  form  A"f  "B'\  Fig.  144,  as  will  correspond  most  closelj- 
to  the  actual  influence  line.  The  criterion  is  then  the  ordinary  one  for  moments,  the  segments 
or  lengths  to  be  used  in  getting  average  load  being  in  any  case  the  distances  corresponding  to 
A"f'  and  f  B" . 

The  position  of  loads  for  maximum  shear  or  web  stress  may  be  found  in  a  manner 
similar  to  that  explained  above.  The  same  criterion  as  for  shear  in  a  square  truss  may 
generally  be  employed,  using  the  truss  length  for  all  panels  except  the  end  ones,  while  for 
these  end  panels  use  the  truss  length  plus  a'A'\  Fig.  144, 


lOO 


MODERN  FRAMED  STRUCTURES. 


It  is  to  be  noted  that  the  above  approximate  methods  deal  only  with  the  positions  of  the 
loads  and  not  with  actual  stresses.  It  is  at  most  a  question  of  which  of  two  or  three  different 
loads  shall  be  at  the  point,  when  any  one  of  them  would  give  very  close  results  In  most 
cases  the  approximate  methods  give  the  correct  wheel  and  hence  correct  results. 


Fig.  145. 


In  getting  stresses  all  loads  between  c  and  i  are  to  be  treated  in  the  same  way  as  in  a 
square  truss.  That  is,  one  half  acts  between  (7  and  /  of  truss  AB,  and  the  other  half  between 
C  and  B'  of  truss  A' B' .  The  loads  from  a  \.o  c  and  from  i  to  b  are  best  treated  by  finding 
the  floor-beam  reactions  at  c  and  i  due  to  these  loads,  and  then  transferring  one  half  of  each 
floor-beam  load  to  each  truss.  The  trusses  are  then  treated  independently  in  the  ordinary 
way. 


TRAIN-LOADS  ON  TRUSS-BRIDGES  OVER  100  EEET  IN  LENGTH.  loi 


CHAPTER  VI. 

CONVENTIONAL   METHODS   OF  TREATING  TRAIN-LOADS  ON  TRUSS-BRIDGES  OVER 

100  FEET  IN  LENGTH.* 

109.  The  Train-load  now  universally  taken  in  the  dimensioning  of  mennbers  in  railway 
bridges  in  America  consists  of  two  of  the  heaviest  engines  in  use  on  tlie  line,  coupled  in  a 
direct  position  at  the  head  of  the  heaviest  known  train-load.  The  weights  of  the  engines  and 
tenders  are  assumed  to  be  concentrated  at  tlie  wheel-bearings,  giving  definite  loads  at  these 
points,  while  the  train-load  is  taken  as  uniformly  distributed.  Although  engine  and  train 
loads  have  greatly  increased  in  the  past  twenty  years,  it  is  highly  probable  that  the  loads  now 
generally  assumed  will  never  be  materially  too  small  for  the  actual  traffic,  except  in  special 
cases.  These  assumed  loads  are  now  equivalent  to  about  4000  lbs.  per  foot  under  the  engines 
and  their  tenders,  or  over  a  distance  of  some  103  feet,  followed  by  a  train-load  of  3000  lbs. 
per  foot.  The  only  exact  solution  for  the  maximum  stresses  produced  by  such  a  load 
moving  across  the  span  has  already  been  given  in  Chapter  V.  If  each  span  had  to  be 
computed  but  once,  there  would  be  no  valid  objection  to  the  use  of  this  exact  method, 
which  has  been  almost  exclusively  employed  in  America  since  about  the  year  1880.  It  has 
come  to  be  a  common  practice,  however,  to  call  for  bids  (either  by  the  span  or  by  the  pound) 
under  general  specifications,  requiring  accurate  "stress  sheets,"  showing  maximum  loads  and 
sizes  of  members,  to  be  submitted  with  each  bid.  If  on  the  average  tliere  are  eight  bidders, 
then  the  computations  have  been  made  independently  eight  times,  and  on  the  average  each 
bidder  computes  eight  spans  for  each  one  he  succeeds  in  building.  When  the  laborious 
method  of  wheel  concentrations  is  employed  the  cost  of  this  high-priced  labor  becomes  a 
considerable  portion  of  the  price  named  for  building  the  bridge.  In  the  end  the  purchaser 
pays  for  the  computations  eight  times  over.  At  present  (1892)  there  is  a  general  desire  on 
the  part  of  tlie  bridge  companies  to  agree  upon  some  conventional  method  of  treating  the 
train-ioad  which  will  lead  to  easy  and  short  computations  without  introducing  material  errors 
in  the  results.  The  consulting  engineers  who  act  for  the  railway  companies,  are  generally 
inclined  to  adhere  to  the  more  exact  methods.  From  a  set  of  observations  taken  on  bridge 
members  under  rapidly  moving  train  loads,  given  in  Engi}iccring  Nczvs,  May  9,  1895,  it 
appears  that  the  actual  stresses,  even  on  the  hip-verticals,  are  not  appreciably  greater  tlian 
those  computed  from  the  same  loads  treated  statically,  and  hence  a  much  greater  weight 
may  be  given  to  the  computed  stresses  than  has  heretofore  been  done,  and  there  is  therefore 
the  greater  reason  to  adhere  to  the  more  rigid  methods  of  analysis.  It  would  probably  be  ad- 
mitted by  all  parties  that  results  differing  by  not  more  than  two  or  three  per  cent  are  practi- 
cally identical  in  bridge  computations,  especially  if  those  differences  are  both  plus  and  minus 
and  do  not  appreciably  change  tlie  total  weight.  It  remains  to  show  that  convenient  methods 
of  solution  can  be  found  which  satisfy  this  requirement  for  truss-bridges  longer  than  100  feet. 

Five  conventional  methods  of  treating  the  train-load  have  been  employed. 

First.  The  use  of  two  concentrated  excess  loads,  placed  fifty  feet  apart,  which  may 
occupy  any  position  in  the  uniform  train  load,  and  which  may  be  conceived  as  rolling  across 
the  span  on  top  of  the  train  load.  They  would  be  near  the  head  of  the  train  for  shears  and 
one  of  them  at  the  joint  in  question  for  moments.f 

*  See  Supplement  to  this  Chapter  on  p.  142. 

t  These  excess  loads  to  be  placed  so  as  to  give  maximum  moments  at  the  several  joints,  the  same  as  described 
below  for  one  concentrated  load. 


I02 


MODERN  FRAMED  STRUCTURES. 


Second.  The  use  of  one  such  concentrated  excess  load  in  place  of  two,  it  always  being  at 
the  joint  in  question,  while  the  train  load  covers  the  whole  span  for  maximum  moments,  and 
reaches  to  a  particular  point  in  the  panel  in  question  for  maximum  shears. 

Third.  The  use  of  a  Uniformly  Distributed  Excess  over  about  one  hundred  feet  at  the 
head  of  the  train. 

Fourth.  The  use  of  what  is  called  an  Equivalent  Uniform  Load. 

Fifth.  The  use  of  a  given  Uniform  Load,  all  of  which  over  a  given  space  in  the  middle  of 
the  train  is  to  be  concentrated  on  two  or  four  axles. 

The  first  of  these  methods  is  apparently  a  near  approximation  to  the  actual  loads,  and  is 
used  by  Prof.  DuBois  in  his  Framed  Structures.  It  is  a  fair  substitute  for  the  wheel  concen- 
trations on  double-intersection  trusses,  but  has  never  come  into  general  use,  the  objection 
being  either  that  it  did  not  sufificiently  shorten  the  work  to  warrant  its  use  in  place  of  the 
more  exact  method,  or  that  no  rational  method  has  been  proposed  for  finding  the  proper 
values  of  the  excess  loads  to  use. 

The  second  method  was  devised  by  Mr.  Geo.  H.  Pegram,  M.  Am.  Soc.  C.  E.,  and 
published  by  him  in  1886.*  It  has  been  introduced  into  some  standard  specifications  because 
of  its  great  simplicity  and  close  agreement  with  the  method  by  wheel  loads.  A  modification 
of  it  is  given  below  in  detail. 

The  third  approximate  method  has  been  used  by  Mr.  C.  L.  Strobel,  M.  Am.  Soc.  C.E., 
for  spans  exceeding  two  hundred  feet,:};  but  has  been  abandoned  by  him  since  train  loads 
per  foot  have  so  nearly  equalled  the  average  load  over  the  engine  portion. 

«The  fourth  method,  of  an  equivalent  uniform  load,  has  been  used  more  or  less  for  many 
years  and  seems  now  likely  to  come  into  more  general  use  and  to  be  incorporated  into  some 
standard  specifications. 

The  fifth  has  been  used  on  the  Norfolk  and  Western  Railroad  since  1889,  and  is  described 
in  Trans.  Am.  Soc.  Civ.  Engrs..  Vol.  XXVI,  p.  149. 

Since  the  method  of  computing  stresses  from  the  actual  wheel-concentrations  will  always 
be  considered  standard,  any  conventional  method  must  use  an  "  equivalent  "  load  of  some 
sort,  and  before  it  will  be  accepted  by  engineers  it  must  be  shown  to  give  practically  equal 
stresses  for  all  main  truss  members  for  all  lengths  of  span  and  panel. 

no.  An  Equivalent  Load  composed  of  a  Uniform  Train  Load  and  one  Moving 
Concentrated  Load. 

Problem. — Given  any  system  of  engine  ivheel  loads  folloived  by  a  uniform  train  load  ( or 
both  preceded  and  folloived  by  such  load),  to  find  zvhat  other  system  of  uniform  train  and  single 
moving  excess  loads  will  give  practically  equivalent  stresses  in  all  members  of  trtisses  over  lOO  feet 
long. 

Take  the  moving  concentrated  load  as  approximately  equal  (in  round  numbers)  to  the 
total  excess  of  the  weight  of  the  engines  and  tenders  over  that  of  the  train  for  the  same  length 
of  wheel  base.    Thus  if 

E  =  total  weight  of  one  engine  and  tender, 
b   =  total  length  of  one  engine  and  tender, 
p'  =  loading  per  foot  of  actual  train, 
Q  =  single  moving  concentrated  load, 
we  have,  for  two  engines  coupled. 

Q^2E  -  2bp'  =  2{E  -  bp').   (i) 

Take  for  the  actual  Q  the  nearest  even  multiple  of  ten  thousand  pounds,  so  that  Q  would 
be,  for  Cooper's  Class  Extra  Heavy  A,  two  engines,  100,000  lbs.  A  considerable  variation 
can  be  made  in  the  value  of  Q  without  materially  affecting  the  results. 


See  Trans.  Am.  Soc.  C.  E.  Vol.  XV,  p.  474;  also  Vol.  XXI,  p.  575. 


f  Id..  Vol.  XXI.  p.  594. 


CONVENTfONAL  METHODS  OF  TREATING  TRAIN  LOADS.  lo^ 


Having  chosen  a  value  tor  Q,  compute  a  corresponding  value  for  p,  the  conventional  train 
loading  per  foot  to  be  used  with  Q,  by  finding  from  the  actual  system  of  wheel  loads  the 
moment  at  the  quarter-point  of  the  length  of  span  to  be  computed  (see  Art.  96),  or  of  a  length 
of  span  nearly  equal  to  the  one  in  question.  Let  /  =  length  of  such  a  span,  and  My^  —  the 
maximum  bending  moment  at  a  point  just  one  fourth  the  length  from  the  end,  regardless 
of  whether  this  falls  at  a  joint  or  not.  This  moment  is  to  be  found  from  a  diagram  such  as 
that  given  on  p.  85,  or  in  any  other  way,  from  the  actual  wheel  loads.  Then  equate  this 
moment  with  the  moment  at  this  point  for  the  "/>  -\-  Q"  loading,  and  obtain 

My,  =  i^pr  +  ^Ql;  (2) 

whence 

32  i%  2<2 

where  /  may  be  taken  in  round  numbers  approximately  equal  to  the  length  of  span  in  question, 
as  the  nearest  multiple  of  ten  feet.    Then  use  this  Q    loading  as  follows: 

For  Maximum  Moments  the  /  loading  must  extend  over  the  entire  span,  and  the  Q  load 
must  be  at  the  centre  of  moments  for  the  member  in  question,  that  is,  it  must  be  at  some 
particular  joint.  If  Q  is  placed  at  the  ;;zth  joint  from  the  left,  counting  the  end  support  zero, 
we  would  then  have  for  the  moment  at  this  joint  from  live  load,  for  a  truss  of  n  equal  panels, 
each  d  feet  in  length, 

^.=^'%-'»)-  (4) 

The  fractional  part  of  this  expression  is  a  constant  for  any  given  truss,  and  hence  these 
moments  can  all  be  taken  out  and  divided  by  the  height  of  the  truss,  thus  _^/t7';/^  <?// r/wni 
stresses  in  a  Pratt  truss  with  one  setting  of  the  slide-rule. 

For  Maximum  Shears  place  Q  at  the  joint  on  the  side  of  the  panel  in  question  from  which 
the  load  is  approaching,  and  let  the  uniform  load  extend  into  this  panel  until  the  maximum 
shear  is  produced.  This  point  is  such  that  a  load  placed  at  this  point  gives  equal  reactions  at 
the  next  forward  joint  and  at  the  forward  end  support.    The  portion  of  this  panel  covered 

with  the  uniform  load  is  then   ^^jrti',*and  the  maximum  shear  in  this  panel  is,  from  live  load. 


5  =  ^ 


pd  {n  —  w)'" 


ft)      in  —  m  \  ^ 


2     («  —  l) 

This  equation  is  also  very  rapidly  evaluated  by  the  slide-rule,  each  term  requiring  but  one 
setting  for  the  whole  span. 

Floor  System  and  Hip-vertical. 

For  computing  the  maximum  moments  and  shears  in  the  stringers  and  floor-beams  and  in 
the  hip-vertical,  which  needs  to  be  done  but  once  for  any  given  span,  the  actual  wheel  loads 
may  be  used,  or  Q  may  be  taken  the  same  as  above,  and  anotlYer  p  found  for  a  length  of  span 
about  equal  to  the  panel  length.  In  this  case /  will  be  found  to  be  negative,  since  the  load  Q 
is  alone  more  than  the  maximum  load  on  the  stringer  in  the  case  of  the  wheel  loads.  It 
would  probably  be  preferable  to  use  the  engine  diagram  for  these  members. 

Note. — The  authors  have  tried  various  expedients  for  obtaining  a  single  set  of  values  of  p  and  Q  to  be 
used  for  all  lengths  of  span  for  any  given  set  of  wheel  and  train  loads,  and  have  made  as  many  as  fifty  com- 
plete s§ts  of  computations  of  stresses  in  all  the  members  of  spans  of  various  lengths  and  for  different  lengths 
of  panel  (which  proves  to  be  a  very  important  consideratic^n;,  comparing  the  results  with  those  obtained  from 

*  See  Art.  69,  Chap.  IV. 


I04 


MODERN  FRAMED  STRUCTORES. 


the  wheel  loads, — all  for  Cooper's  Class  A  loading.  They  do  not  find  that  this  can  be  done  with  satisfactory 
results.  To  use  the  method  with  results  agreeing  closely  with  those  from  the  wheel  loading,  it  seems  to  be 
necessary  to  obtain  the  "/ +  Q"  loading  for  a  length  of  span  approximately  equal  to  that  in  hand. 
Mr.  Pegram  has  advocated  single  values  of  p  and  Q  for  any  given  set  of  wheel  loads  (see  supra),  and  Mr. 
J.  C.  Bland,  M.  Am.  Soc.  C.E.,  has  proposed  the  following  to  correspond  with  Cooper's  four  standard 
loadings : * 

"  Heavy  Grade  "     /  =  4000  ;    Q  =  40000. 

"  Extra  Class  A  "   p  =  3400  ;    Q  —  34000. 

"  Class  A  "  p  =  2800  ;    Q  —  28000. 

"  Class  B  "  /  =  2500  ;    Q  =  25000. 

He  has  made  Q  =  lop  in  every  case,  but  when  p  and  Q  are  found  for  a  particular  length  0/  span,  this 
condition,^  =  ^^oQ,  makes  p  too  large  for  longer  spans,  so  that  the  chord  stresses  become  slightly  too  great. 

How  Specified. 

It  is  important  to  make  the  specification  under  which  bids  are  received  for  building  a 
bridge  perfectly  clear  and  definite.  The  engineer  who  draws  these  specifications,  therefore, 
should  decide  on  a  set  of  values  of  p  and  Q  to  be  used  for  the  particular  length  of  span  and 
for  the  loading  assumed,  and  state  that  such  and  such  lengths  of  span  shall  be  computed  for 
such  and  such  uniformly  distributed  and  rolling  concentrated  loads,  the  web  members  to  be 
computed  for  the  actual  maximum  shear  on  the  panel  by  equation  (5). 


Tabular  Form  for  Computations. 


In  making  the  computations  for  this  conventional  loading  the  work  may  be  conveniently 
arranged  as  follows : 

TOTAL  CHORD  STRESSES— ONE  TRUSS. 


Member. 

(«  —  7n)7n 

Stress. 

Remarks. 

The  quantity  in  the  second  column  is  a  constant  for  all  members  in  one  span.  The  slide- 
rule  is  set  once  for  this  as  a  constant  multiplier  and  all  the  stresses  taken  out.  If  the  dead 
load  is  desired  separately,  the  w  is  to  be  omitted  from  column  2  and  given  a  column  to  itself, 

,  •  ,      ,•       ,      ,   •  d'^^ 
this  neadmg  then  bemg  — y. 

In  computing  the  web  stresses  the  live  and  dead  load  shears  are  computed  separately, 
and  then  combined  as  in  the  following  tabular  form  : 

WEB  STRESSES— ONE  TRUSS. 


Dead  Load  Shear. 

Live  Load  Shear. 

Member. 

w  -{-  I  —  '2tn 

ivd 

X   = 

4 

D.  L.  Shear. 

pd 

n  —  }n 

X 

Q 

Sum  = 

L.  L.  Shear. 

Total 
Shear. 

Secant. 

Stress. 

Member. 

w  —  I 

X — 

4 

n 

2 

I 

2 

3 

4 

5 

6 

7 

8 

9 

ID 

1 1 

12 

The  total  shear  in  the  mth.  panel,  for  the  uniform  load  reaching  into  the  panel  to  give 
maximum  shear,  and  the  concentrated  load  Q  at  the  loaded  panel  joint,  when  w  ■—  dead  load 
per  foot,  is 


Total  shear  on  one  truss 


zvd,                 ,  ,  pd(n  —  mY  ,  Q  (n  —  m\ 
(«  +  I  —  2tn)  +  ^  ^  +  . 


4 


(6) 


*  For  ihe  wlieel  diagrams  for  these  loads  see  p.  79. 


CONVENTIONAL  METHODS  OF  TREATING  TRAIN  LOADS.  105 


These  terms  are  worked  out  in  the  above  tabular  form.  The  results  in  columns  5  and  7 
are  added  to  give  the  live  load  shear  in  column  8,  and  then  these  results  are  added  alge- 
braically to  those  in  column  3  to  obtain  the  total  shear  as  given  in  column  9.  These  are  at 
once  the  stresses  in  the  verticals,  and  when  multiplied  by  the  secant  of  the  angle  the  stresses 
in  the  diagonals  are  found.    These  forms  are  for  the  Pratt  or  Warren  type. 

METHOD  OF  EQUIVALENT  UNIFORM  LOADS. 

III.  Definition,  Equations,  and  Argument. — An  "  equivalent  load  "  is  one  which  would 
produce  the  same  maximum  stresses  in  all  the  members  as  are  caused  by  the  actual  wheel- 
loads.  Evidently  no  single  equivalent  uniform  load  can  be  found  to  do  this.  It  remains  to 
find  a  uniform  load  which  will  give  the  nearest  approximation  to  this  result.  The  moment 
diagram  or  equilibrium  polygon  for  a  uniform  load  on  a  jointed  structure  has  its  vertices  lying 
in  a  parabola  (Art.  50),  while  for  a  plate  girder  it  is  a  parabola.  For  excessive  wheel  concen- 
trations iiear  the  head  of  the  load,  the  polygon  joining  maximum  moment  ordinates  would  be 
below  the  parabola  at  points  on  the  two  sides  ot  the  centre  of  the  span,  somewhat  as  shown 
in  Fig.  146,*  the  full  line  being  the  moment  curve  for  the 
equivalent  uniform  load,  and  the  dotted  line  being  that  V — f  \ 

joining  maximum  moment  ordinates.    These  curves  will  — r  j  1  i     /  / 

cross  each  other  at  about  the  quarter  point.  We  might  \  \  i 
therefore  find  the  equivalent  uniform  load,  so  far  as  the  \  ^| 

chord  stresses  are  concerned,  by  finding  what  uniform  ^vi  \ 

load  will  give  the  same  moment  at  the  quarter  point  ^v^^^" 
which  is  produced  there,  for  that  length  of  span,  by  the  ^^-'llti;^:^ 
actual  wheel  loads.    To  do  this  we  must  first  find  what 
the  maximum  moment  at  this  quarter  point  is  for  any 

given  length  of  span,  for  the  actual  wheel  loads  assumed,  as  described  in  Chap.  V.  (It  is  not 
necessary,  for  this  purpose,  that  this  point  should  fall  at  a  joint.)  Call  this  moment  M^.  It 
now  remains  to  find  the  uniform  load  p  per  foot  which  will  give  this  same  moment  at  this  point. 

For  a  uniform  load  p  over  the  entire  span  of  n  equal  panels,  each  of  length  d,  the  moment 
at  the  quarter  point  is 

M^^j^pi'-^  (7) 

whence 

f=hi^  w 

Having  found  M^,  the  moment  at  the  quarter  point  for  the  actual  wheel  loads,  eq.  (8")  gives 
us  at  once  the  equivalent  uniform  load  which  wil'  produce  the  same  moment  at  this  point. 
Having  found  p,  we  may  write  for  the 

Moment  at  the  ;;/th  joint  —  vt)m,  (9) 

where  the  end  joint  is  called  zero. 

The  proportional  moment  error  in  this  case  will  br  ,ery  small  In  computing  the  shear 
with  equivalent  uniform  live  loads,  it  is  customary  to  treat  the  load  conventionally,  assuming 
that  all  the  joints  on  one  side  of  a  gi^en  panel  are  fully  loaded  and  all  the  joints  on  the  other  side 
wholly  unloaded.  With  this  assumption,  which  cannot  possibly  be  realized,  very  close  agree- 
ments ^can  be  obtained  with  the  rigid  method  from  actual  wheel  loads,  tuheii  the  panel  length 
is  not  less  than  one  eighth  of  the  span;  when  uniform  loads  are  assumed  the  shears  should  be 
always  taken  out  in  this  way.  When  there  are  more  than  eight  panels,  and  when  the  engine 
loading  is  very  much  greater  than  the  train  loading,  the  shears  are  always  too  small. 

*  This  figure  represents  the  moment  diagram,  in  dotted  lines,  for  two  concentrated  loads,  moving  across  the  span 
a  fixed  distance  apart. 


io6 


MODERN  FRAMED  STRUCTURES. 


The  equation  for  maximum  shear,  due  to  both  dead  and  live  load,  for  the  equivalent 
uniform  load  w\  found  for  moments  as  above,  is 

Maximum  shear  in  the  mth  panel  =  S„,  =  —{n  —  2m  -\-  l)  -\-  —J^n  —  m){n  —  m-\-  i),    ( lo) 

where  w  =  dead  load  per  foot, 

p  =  uniform  live  load  per  foot, 
n  =  number  of  panels  in  span, 

m  =  number  of  panels  in  question  counting  from  unloaded  end, 
d  =  panel  length  in  feet. 
For  live  load  shear  only  we  have 

pd 

=  ^i''  +  ^)  (»  0 

These  equations,  (9),  (lo),  and  (i  i),  are  quickly  evaluated  by  the  aid  of  the  slide-rule. 
For  one  truss  take  one  half  the  values  given  by  these  equations. 

112.  Application  to  Systems  with  Inclined  Chords. — For  the  Dead  Load,  whether  this  be 
supposed  to  be  all  concentrated  on  the  bottom  chord  or  partly  at  the  unloaded  joints,  find  all 
the  dead  load  stresses  from  a  single  Maxwell  diagram  as  shown  in  Art.  46,  and  tabulate  these 
stresses  in  all  the  members. 

For  the  Live  Load  Chord  Stresses,  cover  the  entire  span  with  the  equivalent  uniform  live 
load,  and  make  another  diagram  for  these  stresses,  and  tabulate  the  stresses  found  in  the 
chords.  Or  the  better  method  would  be  to  find  these  stresses  from  the  dead-load  stresses 
with  one  setting  of  the  slide-rule. 

For  the  Live  Load  Web  Stresses,*  assume  a  left  end  reaction  of  100,000  lbs.,  treat  the  span 
as  a  cantilever  with  the  left  end  free  and  subjected  to  this  upward  reaction,  and  diagram  the 
stresses  in  all  the  members  for  this  one  reaction  only,  and  tabulate  the  stresses  in  the  web 
members.  Now  compute  the  left  end  reactions  for  successive  positions  of  the  train  load  as  it 
is  backed  off  from  the  left  towards  the  right,  assuming  that  the  joint  at  the  head  of  the  load 
may  be  fully  loaded  and  the  next  one  in  advance  entirely  unloaded.  These  reactions  are  found 
from  the  formula 

where  N  =  number  of  joints  loaded, 

n  =  number  of  panels  in  the  span, 
p  =  equivalent  uniform  load, 
d  =  panel  length. 

Having  these  reactions,  and  the  live  load  web  stresses  for  a  reaction  of  100,000  lbs.,  the 
actual  live  load  web  .stresses  are  found  by  multiplying  those  found  for  the  ioo,000  lb.  reaction 
by  the  actual  reaction,  by  means  of  the  slide-rule. 

The  tabulated  form  would  be  as  follows  : 


Chord  Members. 

Web  Members. 

Mem. 

Dead  Load 
Stress. 

Live  Load 
Stress. 

Total 
Stress. 

Mem. 

Dead  Load 
Stress. 

Stress  for 
/?  =  100,000 

N(N+  I) 

Live  Load 
Stress. 

Total 
Stress. 

I 

2 

3 

4 

5 

6 

7 

8 

9 

10 

II 

Columns  2,  3,  6,  and  7  are  obtained  from  the  diagrams.  Column  10  is  found  by  multiply- 
ing together  the  corresponding  results  in  columns  7  and  9,  and  dividing  by  100,000. 


*  See  also  Art.  80. 


CONVENTIONAL  METHODS  OP  TREATING  TRAIN  LOADS. 


Columns  9  and  lO  are  found  by  the  slide-rule — -the  former  from  one  setting,  the  latter  by 
separate  settings  for  each  result. 

Nothing  would  be  gained  by  obtaining  the  sum  of  the  dead  and  live  load  stresses  from 
one  diagram  drawn  for  the  sum  of  the  unit  loads,  since  the  dead  load  web  stresses  must  be 
taken  out  separately. 

By  this  method  all  the  maximum  stresses  in  the  main  truss  members  of  a  large  bridge 
can  be  found  in  a  few  hours.  The  results  by  diagram  should  always  be  checked  by  computing 
one  or  two  of  the  simpler  cases.  Then  since  the  diagram  checks  itself  it  may  be  assumed 
that  if  one  part  is  right  it  is  all  right. 

ii2«.  Accuracy  of  Results  by  the  Conventional  Methods. — The  following  table 
embodies  the  results  of  computations  by  the  two  conventional  methods  here  proposed,  as 
compared  with  the  rigid  results  from  the  actual  wheel  loads,  all  for  Cooper's  class  "  Extra 
Heavy  A"  loading,  as  given  on  p.  79.  For  engines  very  much  heavier  than  the  train  load  no 
satisfactory  convention  has  been  found  for  plate  girders,  stringers,  floor-beams,  and  hip- 
verticals.    For  these  the  actual  wheel  loads  should  be  employed. 

A  careful  study  of  this  table,  which  embodies  the  best  results  the  authors  could  obtain  for 
an  engine  load  approximating  4000  lbs.  per  foot  for  100  feet  at  the  head  of  a  train  load  of  3000 
lbs.  per  foot,  will  show  : 

I.  That  the  "/  +  <2  "  loading  gives  better  results  for  the  web  system  than  the  equivalent 
uniform  load.    The  results  on  the  chord  members  are  identical,*  since  the      -\-  Q"  loading 


2.  That  the  equivalent  uniform  load  gives  values  very  much  too  small  for  the  web  system 
when  the  number  of  panels  is  greater  than  eight.  If  this  loading  were  treated  rigidly  for 
shear,  the  discrepancy  would  be  much  greater.  This  shows  that  if  equivalent  loads  are  used, 
a  different  equivalent  must  be  used  for  shears  from  that  which  is  used  for  moments. 

3.  That  the  +  Q"  loading  gives  very  nearly  correct  values  for  both  chord  and  web 
members,  except  for  the  counters  where  the  result  is  always  large  by  from  six  per  cent  in  the 
3C)0-foot  span  to  43  percent  in  the  150-foot  span.  If  this  method  should  be  employed,  then 
the  same  cojnpiitcd  fibre  stress  per  square  inch  could  be  allowed  in  the  counters  as  in  the 
mains.  When  these  members  are  rigidly  computed  a  liberal  reduction  in  the  working  stress 
is  made  for  "impact." 

4.  That  since  the  facilities  and  aids  to  wheel-load  computation  are  now  so  simple  and 
efficient,  and  since  this  is  acknowledged  to  most  nearly  represent  the  actual  conditions  of 
service,  such  loads  should  continue  to  be  used  in  the  actual  dimensioning  of  both  truss  and 
plate  girder  simple  span  bridges  when  they  are  prescribed  in  the  specifications. 

5.  That  for  estimating  cross-sections  and  weights  of  bridges  for  the  purpose  of  bidding  on 
work,  the  +  (2"  loading  might  well  be  employed.  The  total  weight  found  from  its  use 
would  probably  not  differ  by  more  than  one  or  two  tenths  of  one  per  cent  from  those  found 
from  the  rigid  method.  The  final  sections,  however,  should  be  fixed  by  the  wheel-load 
method  of  computation. 

*  The  discrepancies  in  the  table  are  all  in  the  last  significant  figure,  due  to  neglecting  the  remaining  figures. 
Note. — A  Supplemental  Note,  added  to  this  Chapter  in  the  Third  Edition  of  this  Work,  will  be  found  on  p.  142. 


is  equal  to  an  equivalent  uniform  load  oi  [p  -\- 


per  foot  for  moments. 


io8 


MODERN  FRAMED  STRUCTURES. 


COMPARATIVE  RESULTS  OF  TRUSS  COMPUTATIONS  BY  RIGID  AND  CONVENTIONAL  METHODS 


BCDE  FGHl  K 


Stresses  in  Thousands  of  Pounds. 


span,  Panel 

(«/) 

Wheel 

Loads. 

(  /  +  0  Load.    Eqs.  (4)  and  (5). 

(w')  Equivalent  Uniform  Load. 
Eos.  (q)  and  (iiV 

Members. 

Dead- 

Lengths, 

load 

Live  Load 

Total 

Live  Load 

Total 

Ratio  to 

Live  Load 

Total 

Ratio  to 



Stresses. 



Stresses. 

Stresses. 

Stresses. 

Stresses. 

Wh.  Lds. 

Stresses. 

Stresses. 

Wh  Lds. 

ab~bc 

20.2 

67 

5 

87 

66 

9 

87 

I 



99.6 

66 

9 

87 

1 



99.6 

/  =  icq' 

BC-cd 

30.2 

94 

8 

127 

2 

100.3 

130 

5 

loi .  5 

100 

3 

130 

5 

101 . 5 

d  =  20' 

CD 

30.2 

97 

0 

127 

2 

100 

3 

130 

5 

loi .  5 

100 

3 

130 

5 

lOI  .  5 

w  —  1.25 

■  '    0    T  $^ 

Jf            ^.  L  0 

aB 

32 . 2 

117 

9 

150 

I 

106 

5 

138 

7 

92.4 

106 

8 

139- 

0 

92 . 0 

^  —  100 

Ur 

JjC 

T 

1 D .  1 

62 

2 

78-3 

69.7 

85 

8 

109  ■  6 

64 

I 

80 

2 

102 . 4 

to  =  4.  ^8 

0.0 

29 

2 

29 

2 

39 

c 
3 

39 

5 

135  ■  2 

32 

2 

32 

2 

110.3 

/  —  150' 

ab—bc 

42.0 

IIO.O 

152. 

0 

108 

, 

150 

I 

98.8 

108 

5 

150 

5 

99  0 

BC-cd 

67.2 

169.5 

236.7 

172 

9 

240 

I 

EOI  .4 

173 

6 

240 

8 

101.7 

d  =  25' 

CD 

75.6 

185.9 

261 

5 

194 

5 

270 

I 

103  .  3 

195 

3 

270.9 

103 .6 

IV  =  1.50 

aB 

63  .0 

166. 

5 

229 

5 

162 

I 

225 

I 

98.1 

162 

3 

225 

5 

98.2 

Be 

37.8 

1 10 

8 

148.6 

112 

6 

150 

4 

lOI  .2 

108 

2 

146 

0 

98. 2 

Q  =  100 

Cd 

12.6 

62 

6 

75 

2 

71 

8 

84.4 

112. 2 

64.9 

77 

5 

103.0 

w'  =  3.88 

De* 

12.6 

3' 

18 

6 

39 

3 

26 

7 

32 

5 

19 

9 

107 .0 

ab-hc 

56.3 

122 

7 

179 

0 

117 

6 

173 

9 

97-2 

117 

6 

173 

9 

97.2 

/  =  200' 

BC-cd 

100.0 

211 

5 

3" 

5 

209 

I 

309 

I 

99.2 

209 

I 

309 

I 

99-2 

d  =  20' 

CD-d^ 

272 

3 

403 

6 

274 

5 

405 

8 

100.5 

274 

5 

405 

8 

100.5 

DE-ef 

150.0 

312 

7 

462 

7 

313 

7 

463 

7 

100.2 

3'3 

7 

463 

7 

100. 2 

w  =  1.75 

EF 

322. 

6 

478 

9 

326 

8 

•483 

I 

100.9 

326 

8 

483 

I 

100.9 

/  =  2.66 

Q  =  100 

aB 

96.8 

209 

9 

306 

7 

202 

4 

299 

2 

97-6 

202 

5 

299 

3 

97-6 

tv'  =  3.66 

Be 

75 . 3 

170 

5 

245 

8 

165 

4 

240 

7 

97-9 

162 

0 

237 

3 

96.7 

Cd 

53-8 

133 

8 

187 

6 

132 

0 

185 

8 

99.0 

126 

0 

179 

8 

95.8 

De 

32.3 

100 

8 

133 

I 

102 

2 

134 

5 

lOI  .  0 

94 

5 

126 

8 

y  3  -  J 

Ef 

10.8 

72 

0 

82 

8 

76 

I 

86 

9 

104 . 9 

67 

5 

78 

3 

94 . 6 

Fcr  * 

in  R 

47 

2 

•  36 

4 

53 

6 

42 

8 

117  6 

45 

0 

34 

2 

94  •  0 

ab,  be 

60.  2 

129 

7 

189 

9 

125 

I 

185-3 

97.6 

125 

3 

185 

5 

97-7 

/    '  200' 

BC,  cd 

103.  I 

214 

3 

317 

4 

214 

4 

317 

5 

100.0 

214 

8 

317 

9 

100. 2 

=  25' 

CD,  de 

128.9 

266 

7 

395 

6 

268 

0 

396.9 

100.3 

268 

5 

397 

4 

100.4 

w  =  1.75 

DE 

137-5 

282 

2 

419 

7 

285.9 

423 

4 

100.9 

286 

4 

423 

9 

lOI  .0 

*  =  2.66 

^     :  100 

aB 

07.  7 

210 

7 

308.4 

203 

2 

300.9 

97-6 

203 

3 

301 

0 

97-6 

w'  =  3.66 

Be 

69.8 

159 

2 

229 

0 

156 

I 

225 

9 

98.7 

152 

5 

222 

3 

96.2 

Cd 

4.1 .0 

114 

7 

156.6 

115 .0 

156.9 

100.2 

108 

9 

150-8 

96-3 

De 

14.0 

76.3 

90 

3 

79-9 

93 

9 

104.0 

72 

6 

86 

6 

95-9 

Ef* 

14.0 

43 

6 

29 

6 

51 

2 

37 

2 

125 .0 

43 

6 

29 

6 

100 . 0 

/  =  250' 

ab,  be 

87.  Q 

161 

3 

249 

2 

154 

7 

242 

6 

97.4 

154 

8 

242 

7 

97-4 

BC,  ed 

156.2 

277 

4 

433 

6 

275 

0 

431 

2 

99-4 

275 

2 

431 

4 

99-4 

d  —  25' 

CD,  de 

.  I 

361 

3 

566 

4 

361 

0 

566 

I 

100.0 

361 

2 

566 

3 

100.0 

tt/  =  2.00 

DE,  ef 

*  J-T  •  *-r 

409 

2 

643 

6 

412 

5 

646 

9 

100.5 

412 

8 

647 

2 

100.6 

^  =  2.72 
Q  =  100 

EF 

21A .  I 

416 

4 

660 

5 

429 

7 

673 

8 

102.0 

430 

0 

674 

I 

102. 1 

aB 

142. 8 

261 

4 

404 

2 

251 

3 

394 

I 

97-5 

251 

I 

393 

9 

97-5 

Be 

1 1 1 . 0 

210 

9 

321 

9 

204 

2 

315 

2 

98.1 

200 

9 

311 

9 

96.9 

Cd 

7Q . 

165 

7 

245 

0 

161 

9 

241 

2 

98-5 

156 

2 

235 

5 

96.1 

De 

4.7  .6 

125 

I 

172 

7 

124 

4 

172 

0 

99-6 

117 

2 

164 

8 

95-4 

Ef 

J  .Q 
J  •  V 

89 

6 

105 

5 

91 

6 

107 

5 

101.9 

83 

7 

99 

6 

94.4 

Fg* 

15-9 

59 

3 

43 

4 

63 

7 

47 

8 

no. I 

55 

8 

39 

9 

91.9 

ab,  be 

120. 1 

189 

2 

309 

3 

183 

7 

303 

8 

98.2 

183 

6 

303 

7 

98.2 

/  =  300' 

BC,  eJ 

213-5 

326 

6 

540 

I 

326 

5 

540 

0 

100.0 

326 

4 

539 

9 

99-9 

=  30' 

CD,  de 

280.5 

426 

0 

706 

5 

428 

6 

709 

I 

100.4 

428 

4 

70S 

9 

100.3 

DE,  ef 

320.2 

476 

8 

797 

0 

489 

8 

8io 

0 

loi .  6 

489 

6 

809 

8 

lOI  .  5 

w  =  2.25 

EF 

333-6 

488 

4 

822 

0 

510 

2 

843 

8 

102 . 7 

510 

0 

843 

6 

102.6 

/  =  2.78 

98.0 

Q  =  100 

aB 

193.8 

305 

.2 

499 

0 

296 

7 

490 

5 

98.3 

295 

2 

489 

0 

tv'  =  3.44 

Be 

150.7 

246 

2 

396 

9 

240 

I 

390 

8 

98.5 

236 

2 

386 

9 

97-5 

Cd 

107.6 

192 

■9 

300 

5 

189 

4 

297 

0 

98.  S 

183 

7 

291 

3 

96-9 

De 

64.6 

145 

9 

210 

5 

144 

6 

209 

2 

99.4 

137 

8 

202 

4 

96. 1 

Ef 

21.5 

105 

.0 

126 

5 

105 

7 

127 

2 

100.5 

98 

4 

119 

9 

94.8 

Eg* 

21.5 

69 

-7 

48 

.2 

72 

7 

51 

2 

106.2 

65 

6 

44 

I 

91.5 

Note. — The  heights  of  these  spans  were  as  follows  :  For  100'  span,  A  =  25';  for  150'  and  200'  spans  (10  panels), 
h  =  2%'  ;  for  200'  (8  panels)  and  250'  spans,  h  =  32';  for  300'  span,  h  =  38'.    The  loads  given  in  the  first  column  are 


for  two  trusses. 


CONVENTIONAL  METHODS  OF  TREATING  TRAIN-LOADS.  xoZa 


112b.  Cooper's  Conventional  Wheel  Loads. — In  Vol.  XXXI  of  Trans.  Am.  Soc.  C.  E., 
p.  174  (Feb.  1894),  Mr.  Theodore  Cooper,  M.  Am.  Soc.  C.E.,  recommends  the  following 
system  of  engine  and  train  loads  for  all  bridge  computations,  the  resulting  stresses  to  be  multi- 
plied by  a  constant  factor  (by  slide-rule)  to  reduce  them  to  their  equivalents  for  any  other 
system  of  engine  and  train  loads  : 

6        OOQO  oooo          o        PQoo  0000 

g        oooo  oooo          5        OQoo  0000 

§        oooo  oooo  o         q     01     o.     °  0060 


2,500  LBS. 
PER  FT. 


o        S     8    S     "         2  12  o        S     S    S     S         in     uf      lo-  12 

o  oooo  on  00  n  OOOO  no  no 

*t- 7'5->^5'-x-5V  5'-^    9'    »*-5'-»«-  e'^s'^  8'  ^  T^B  ^S'^  5'-*^5'*^     9'   -m^  5'-*^  6'-^5'->^5^ 
E  25 

LOAD  ON  BOTH  RAILS. 

By  using  this  system  of  loads  as  representing  the  least  allowable  loads  for  any  railroad 
bridge,  and  finding  the  corresponding  live-load  stresses,  we  could  then  multiply  these  by 
factors,  as  1.2,  1.4,  1.6,  etc.,  so  as  to  change  the  stresses  to  those  arising  from  any  correspond- 
ingly increased  loads.  Thus,  when  the  factor  1.6  is  used,  we  shall  have  40,000  pounds  on  the 
driver  axles,  in  place  of  25,000  pounds.  Hence  we  could  call  this  typical  engine  "  E  40  "  (the 
one  used  primarily  being  typical  engine  "  E  25  "),  and  the  loads  would  become 

o        Ooop  oooo         o        oooo  oooo 

X        opoo  oooo         6        oooo  oooo 

o        oooQ  oooo  o         oooo  oooo 


4,000  LBS. 
PER  FT. 


oooo  •"t't^^t  (o'         oooo  ^'■*Tf^' 

^  Tf't^^  (N(N(N(N  't'^'t^  (NCMCMCN 

o  oooo  oooo  o  OOOO  oooo 

7'.5i»-»  5'**  5'-^  5' >*-    9'    -K- 5' **-  6'  ^  5' ^    S'  7'5  ■»*- 5'***-  &'->*-  S'x-    ^    -«  5'*<-  6' 5'-k-5'- 

LOAD  ON  BOTH  RAILS. 

This  loading  corresponds  very  closely  to  that  of  the  heaviest  Lehigh  Valley  engines. 
The  live-load  stresses  for  this  loading  would  be  obtained  by  multiplying  all  the  stresses  found 
for  the  '  E  25  "  loading  by  multiplying  them  all  by  1.6.  This  could  be  done  with  a  single 
setting  of  the  slide-rule. 

This  method  of  treating  train-loads  has  now  (1897)  come  into  common  use.  For  the 
purpose  of  utilizing  this  very  satisfactory  system,  the  moment  table  on  page  lo8<^  has  been 
prepared.*  It  is  less  elaborate  than  that  on  page  79,  but  it  serves  the  same  purpose.  It  is 
computed  for  "Class  E  30"  loading,  this  being  the  mean  of  the  two  classes  shown  above. 
These  moments  are  given  in  lOOO-lb.  units,  and  for  one  rail,  or  for  one  half  the  live  load.  For 
any  other  class,  as  for  "  Class  E  27,"  take  |^  of  the  stresses  found  by  the  use  of  the  table. 
The  particular  wheel  to  put  at  any  joint  is  found  as  described  in  Chapter  V.  By  comparing 
the  table  on  p.  loZb  with  that  on  page  79  it  will  be  found  that  the  moments  given  in  the 
former  in  column  ^  correspond  to  those  given  in  columns  10  and  i  of  the  latter,  reading  from 
the  bottom  upwards  in  this  case.  As  these  are  the  only  parts  of  the  larger  table  which  are 
used  for  finding  abutment  reactions,  and  since  the  panel-load  concentrations  can  be  obtained 
by  summing  the  proper  moment-increments  in  column  /,  it  would  seem  that  the  smaller  table, 
as  here  given,  were  the  more  convenient.  This  table  also  serves  to  include  the  moment  from 
uniform  loads  to  a  distance  of  324  feet  from  the  first  wheel  of  the  head  engine.  The  prepa- 
ration of  the  table  is  evident  from  the  headings.  The  moments  given  are  the  moments  of 
the  loads  on  one  rail,  while  the  loads  given  in  the  diagrams  above  are  the  total  wheel  loads 
on  both  rails.  Each  horizontal  line  of  the  table  applies  to  that  particular  wheel  or  point  in 
the  train  loading.  The  figures  set  in  the  upper  part  of  the  space  are  used  only  in  compiling 
the  other  results  in  the  table. 


*  After  one  prepared  by  Mr.  C.  L.  Strobel,  and  contributed  to  the  R.  R.  Gazette  of  Dec.  26,  1896,  by  Mr.  T.  L. 
Condron. 


MODERN  FRAMED  STRUCTURES. 


COOPER'S  E  80-MOMENT    TABLE   FOR  TWO  106i-TON   LOCOMOTIVES,  FOLLOWED  BY  3000  LBS. 

PER   LINEAR   FOOT   OF  TRACK. 
(loads  in  thousands  of  pounds,    moments  in  thousands  of  foot-pounds.) 

For  One  Rail. 


No.  of 
Wheel 

or 
Load. 

b 

Distance 

from 
istWheel. 

c 

Wheel 
Load. 

d 

Sum 
of 
Loads. 

e 

Dist'ce 

be- 
tween 
Wheels 

/ 
Incre- 
ment of 
Moment. 

1  * 
Moment. 

1 

0 

7-5 

7.5 

0 

J 

8 

15 

33.. 5 

8 

60 

60 

3 

o 

13 

'5 

37.5 

S 

112. 5 

173.5 

4 

o 

18 

'5 

53.5 

5 

187.5 

360 

5 

o 

23 

15 

67.5 

5 

262.5 

633.5 

6 

o 

33 

9-75 

77.35 

9 

607.5 

1230 

1 

o 

37 

9-75 

87 

5 

386.25 

1616.25 

8 

o 

43 

9-75 

96,75 

6 

522 

3138.35 

9 

o 

48 

9-75 

106  5 

5 

483-75 

3633 

10 

J 

66 

7-5 

114 

8 

852 

3474 

11 

o 

64 

IS 

139 

8 

gi2" 

4386 

13 

o 

69 

15 

144 

5 

645 

503 1 

13 

o 

74 

■5 

159 

5 

720 

5751 

14 

o 

79 

1 5 

174 

5 

795 

6546 

15 

o 

88 

9-75 

183.75 

9 

1566 

8112 

16 

o 

93 

9-75 

193.5 

5 

918.75 

9030.75 

17 

o 

99 

9-75 

203.25 

6 

T161 

10191 .75 

18 

o 

104 

9-75 

313 

5 

10x6.25 

11208 

A 

\ 

X 

X 

A 

X 

\\           b  c 
V  .50'  > 

d  e 
<  1.50-  

No.  of 
Wheel 

or 
Load. 


19 
20 

31 
32 
23 
24 
25 
36 
87 
38 
39 
30 
31 
33 
33 
34 
35 
36 
37 
38 
39 
40 


Distance 

from 
ist  Wheel. 


114 


Wheel 
Load. 


124 


134 
144 


154 


164  I 
174  I 
184  i 

194  I 

i 

204  j 

214 

324 

234 

344 


254 
264 
374 
384 
394 
304 
314 
324 


Sum 
of 
Loads. 


228 


243 


258 
273 


288 


303 


318 


333 


348 


363 
378 


393 


408 


423 


438 
453 


468 


483 


498 

513 


538 


Dist'ce 

be- 
tween 
Wheels. 


/     I  e 
Incre-  1 

ment  of  ;  Moment. 
Moment.  ! 


13338 


2280 


15618 


2580 


3180 


3330 


18048 


20628 


23358 


26238 


29268 


3480 
3630 


32448 
35778 


3780 


3930 


39258 
42888 
46668 
60598 


4080 


54678 


4380 


4680 


:  59908 

I 

'  63288 
67818 
'  72498 


4830 


4980 


77328 


82308 
87438 


92718 


To  illustrate  how  the  table  is  used,  the  following  example  is  given  of  a  200-ft.  span  of  eight  panels. 

With  wheel  4  at  <r  we  have  : 

Length  of  bridge  to  right  of  <.     150  ft. 


train  to  left  of  4  (from  table) 


Total  length  of  train  on  bridge   

Nearest  point  of  train  load  in  table  (load  24)  

Distance  from  load  point  24  to  end  of  truss. 
Total  weight  of  train  to  point  24  (from  table) . . 


168  " 
164  " 


303  lbs  (thousands) 


Increment  of  moment  to  be  added    1,112  ft.-U)- 

Moment  of  entire  train  about  point  24  (from  table)   ..  26,238 

Increment  of  moment  as  above      1,212 


Total  moment   27.450  X  5  =  6,562 

Negative  moment  about  wheel  4    360 


Required  moment  at  c  (thousands  of  ft. -lbs.). 


LATERAL  TRUSS  SYSTEMS.  109 


CHAPTER  VII. 

LATERAL  TRUSS  SYSTEMS. 

113.  The  lateral  pressure  upon  a  bridge  or  roof  truss  arising  from  wind  or  the  centrifugal 
force  due  to  loads  moving  in  a  curve,  are  resisted  by  means  of  lateral  horizontal  trusses 
placed  between  the  chords  of  the  vertical  trusses.  The  chords  of  the  vertical  trusses  thus 
form  the  chords  of  the  lateral  trusses.  Roof  trusses  are  usually  braced  laterally  in  pairs,  the 
end  pair  taking  the  greater  part  of  the  end  pressure,  the  others  being  merely  stiffened  against 
buckling,  according  to  the  judgment  of  the  engineer.  The  stresses  in  the  end  lateral  system 
are  computed  in  the  same  way  as  are  those  in  the  lateral  systems  of  bridge  trusses,  which  will 
be  discussed  in  detail. 

The  lateral  systems  of  a  bridge  are  either  of  the  Howe,  Pratt,  or  Warren  type,  cor- 
responding usually  with  the  type  of  the  main  trusses.  Fig.  147  shows  the  upper  and  Fig 
149  the  lower  lateral  system  of  a  through  Pratt  truss.  Fig.  148.  The  wind  pressure  may  be 
from  either  direction,  hence  both  sets  of  diagonals  are  required  throughout.    The  floor-beams 

C  D' 


Fig.  149.  Fig.  150. 


usually  form  the  transverse  members  of  the  lower  lateral  system  in  a  through  bridge,  and  of 
the  upper  lateral  system  in  a  deck  bridge.  The  loads  upon  the  upper  laterals  of  Fig.  148,  or 
upon  the  lower  laterals  of  a  deck  bridge  supported  as  in  Fig.  150,  are  carried  by  them  to  the 
points  C  and  D  or  C  and  D' ,  and  are  then  transferred  to  the  abutments  at  A  and  B  by  means 
o[  portal  bracing  in  the  transverse  planes  of  AC  and  BD.  Two  forms  of  such  bracing  are 
shown  in  Fig.  148  (^^)  and  Fig.  150  {a). 

STRESSES  DUE  TO  WIND  PRESSURE, 

114.  Upper  and  Lower  Laterals. — The  wind  pressure  upon  highway  bridges  is  usually 

taken  at  about  30  lbs.  per  square  foot.  The  exposed  area  is  found  by  adding  the  area  of  the 
vertical  projection  of  the  floor  system  to  twice  the  vertical  projection  of  one  truss,  thus 
assuming  no  live  load  upon  the  bridge  at  the  time  of  maximum  wind  pressure.  The  pressure 
upon  the  upper  half  of  the  truss  is  assumed  to  be  taken  by  the  upper  laterals,  and  that  upon 
the  lower  half  by  the  lower  laterals. 

Instead  of  the  above  loads,  specifications  sometimes  adopt  a  pressure  of  from  lOO  lbs.  to 
150  lbs.  per  lineal  foot  upon  the  unloaded  chord  and  200  lbs.  to  250  lbs.  per  foot  upon  the 
loaded  chord.    The  wind  loads  upon  the  unloaded  bridge  are  usually  treated  as  fixed  loads. 


no 


MODERN  FRAMED  STRUCTURES. 


For  railroad  bridges,  a  pressure  of  from  30  lbs.  to  50  lbs.  per  square  foot  is  assumed  for 
the  unloaded  bridge,  and  about  30  lbs.  upon  bridge  and  train,  the  load  due  to  the  pressure 
upon  the  train  being  treated  as  a  moving  load.  Or,  as  in  Cooper's  specifications,  a  total 
pressure  of  450  lbs.  per  lineal  foot  upon  the  loaded  chord,  300  lbs.  of  which  is  treated  as  a 
moving  load  ;  and  150  lbs.  per  foot  upon  the  unloaded  chord. 

The  adopted  unit  pressures  of  45  lbs.  and  30  lbs.  of  Chap.  Ill  agree  very  well  with  aver- 
age practice  as  regards  bridges. 

The  transference  of  the  pressure  upon  the  train  depends  upon  the  disposition  of  the 
floor-beams  and  lateral  systems,  but  usually  it  may  be  considered  as  being  all  taken  up  by  the 
lateral  system  belonging  to  the  loaded  chord.  This  question,  together  with  the  effect  of  the 
pressure  upon  the  train  not  being  applied  in  the  plane  of  the  floor-beams,  will  be  discussed 
under  the  subject  of  centrifugal  force. 

In  finding  stresses,  loads  on  the  lateral  system  of  the  unloaded  chord  may  be  considered 
as  applied  equally  upon  the  two  sides,  windward  and  leeward  ;  and  for  the  laterals  of  the 
loaded  chord  they  may  be  taken  as  applied  wholly  on  the  windward  side.  The  stresses  are 
then  readily  found  by  methods  already  familiar. 

The  resulting  chord  stresses  should  be  combined  with  those  due  to  dead  and  live  loads  if 
the  live  load  is  considered  as  acting  simultaneously  with  the  wind  load ;  or  if  the  live  load  is 
considered  as  not  so  acting,  then  they  should  be  added  to  the  dead  load  stresses.  The 
stress  in  the  windward  lower  chord  is  thus  often  reversed.  There  is  also,  as  will  be  seen 
later,  a  uniform  compressive  stress  in  the  windward  chord  induced  by  the  action  of  the  portal 
bracing,  and  this  stress  is  to  be  combined  with  the  two  above.  Wherever  a  reversal  of  stress 
occurs,  reliance  must  be  placed  on  the  stringers  to  resist  buckling ;  or  where  this  cannot  be 
done,  as  in  the  end  panel,  the  chord  must  be  counterbraced. 

Where  the  lateral  rods  of  the  system  belonging  to  the  loaded  chord  are  attached  to  the 
flanges  of  the  floor-beams,  the  stresses  thus  caused  in  the  flanges  must  be  taken  into  account 
when  they  are  of  the  same  sign  as  those  caused  by  the  bending  moment. 

Example  i.  Find  the  stresses  of  the  lateral  systems  of  Figs.  148  and  150,  where  AB  =  120  ft.  and 
CC  =  17  ft.  Pratt  system  of  laterals.  Pressure  on  CD  =  100  lbs.  per  foot,  and  upon  A£  =  200  lbs.  per 
foot;  all  to  be  taken  as  a  moving  load. 

Example  2.  Find  the  stresses  in  the  lower  laterals,  Fig.  148,  for  a  pressure  of  150  lbs.  fixed  and  300 
lbs.  moving  load  per  foot.  The  diagonals  are  angle-irons  riveted  to  stringers  and  floor-beams  and  are 
assumed  to  resist  in  tension  and  compression  equally. 

115.  Portal  Bracing. — Form  i.  A  common  form  of  portal  bracing  is  shown  in  Fig.  151. 
£  Z  c  R  '^^^  distance  AC,  —  c,  is  the  length  of  the  end  post  whether  vertical  or 
inclined.  With  the  usual  Pratt  type  of  upper  laterals,  one  half  the  entire 
load  upon  the  intermediate  panels  is  transferred  to  the  point  C,  as  one 
abutment  of  the  lateral  truss.  Call  this  load  R.  There  is  in  addition 
about  one-half  a  full  panel  load  applied  at  C,  and  one-half  at  C  ;  these 
are  due  to  the  pressure  upon  the  end  panel  of  the  upper  chord  and  upon 

P 

the  end  posts.    Call  each  --.    These  external  forces  are  held  in  equilib- 

FiG.  151.  Hum  by  other  unknown  external  forces,  in  the  plane  of  the  portal,  applied 

at  A  and  A' .    Let  the  components  of  these  forces  be  as  represented  in  the  figure. 
From  the  conditions  of  equilibrium  we  have  readily, 

H^H'  =  R  +  P,    and    V  =  -  V  =  {R -\- P)y    .    .    ,    .    ,  (l) 

Assuming  H=  H',  as  is  admissible,  we  have 

H=H'  =  ^.  (2) 


t 

e 
1 

im 

D 

<  b  

V 

LATERAL   J-JtUSS  SYSTEMS. 


Ill 


Passing  the  section  hn  and  taking  centre  of  moments  at  D,  the  piece  CD'  not  being  in 
action  with  wind  from  right,  we  have 

compressive  stress  m  66   —  ^  =:  I — ^ — j~e  ~2 

If  the  upper  laterals  are  of  the  Howe  type,  transferring  their  loads  to  C ,  the  stress  in 
CC  will  be  less  than  that  above  by  the  amount  R. 
From  '2  vert.  comp.  =  o  we  have 

tensile  stress  in  CD  —  V  sec  6  —  {^R  -\-  P)-^  sec  0  (4) 

From  2  mom.  —  o,  centre  of  moments  at  C,  we  have 

compressive  stress  m  DD  =   =   (5) 

The  force  V'  produces  a  uniform  compressive  stress  in  the  post  A'C,  and  the  force  V 
reduces  the  compressive  stress  in  the  portion  AD  of  the  post  AC.  Above  D  there  is  no 
change. 

R-^  P 

The  bending  moment  at  D  and  D  =  H  X  (c  —  e)  =  —  (c  —  e),  and  is  the  greatest 

moment  in  the  posts.  The  maximum  compressive  fibre  stress  in  the  posts  occurs  on  the  inner 
side  of  the  leeward  post  at  D',  where  the  flange  stress  due  to  the  bending  moment  adds  to 
the  direct  stresses  due  to  V  and  to  live  and  dead  loads.  Each  post  should  then  be  designed 
to  resist  this  combined  stress. 

If  the  posts  are  inclined,  the  horizontal  components  of  F  and  V',{Vsin  a),  (Fig,  152),  act 
directly  upon  the  lower  chord  as  a  relief  on  one  side  and  an  addi-  c' 
tional  tension  on  the  other.    These  stresses  are  uniform  throughout  \    y\  y\ 


the  length  of  the  bridge,  and  are  to  be  combined  with  those  due  to 
the  lower  lateral  system  and  to  live  and  dead  loads.    A  reversal  of  Fig.  152. 

stress  often  occurs  in  the  end  panels  of  the  windward  chord,  as  stated  in  the  previous  article. 

A  ?Xr\x\.,  AA' ,\Vl  Fig.  151,  is  usually  inserted  to  distribute  the  pressures  to  the  two 
supports.  At  the  fixed  end  of  the  span  the  stress  in  this  strut  may  be  taken  equal  to  one- 
half  the  load  brought  to  A  by  the  lower  laterals  i^R').    At  the  free  end  it  is  equal  to 

\R\   (6) 

or  to 

+  Fcos^),  (6') 

whichever  is  the  greater.     In  eq.  (6'),  \P'  —  pressure  upon  the  end   panel  of  the  lower 

laterals  applied  directly  at  yl  ;  W  —  total  dead  load  ;  and  V  cos  a  —  upward  pull  on  shoe 

of  windward  post.    The  first  parenthesis  is  thus  the  total  lateral  pressure  upon  A,  and  the 

second  is  the  frictional  resistance  of  the  post,  taking  \  as  the  coefificient  of  friction.  The 

difference  must  be  transmitted  to  A' . 

c  c  W 

When  —  <  Fcos  a,  the  bridge  must  be  anchored  down. 


ForjH  2.   The  portal  of  a  deck  bridge  is  of  the  form  shown  in  Fig.  153. 
The  stresses  in  CC ,  AC,  AA' ,  and  the  direct  stress  in  A'C  are  found  as  in  the 
previous  case.    There  is  no  bending  moment  in  any  member,  nor  is  there  any 
'53-        tension  in  v^C, 


112 


MODERN  FRAMED  STRUCTURES. 


Form  3.  Fig.  154  shows  a  common  form  of  portal.    The  stresses  in  the  main  posts  are 
p  p  as  in  Fig.  151,  the  dangerous  section  being  at  D' . 


H' 


F  >- 

K1- 

TT 

\  1 

E 

m 

■<  X-—' 

i 

c 

1 

a' 

-h  ' 

a| 

'v' 

H 

V 

2  Treating  the  portion  CDD'C  as  a  beam,  we  may  find  the  stress  at 

any  point  F  o{  the  flange  CC,  by  passing  a  section  FE  and  then  taking 
moments  of  the  forces  on  the  right,  about  E.  We  have  then,  from  eqs. 
(I)  and  (2), 

H{c-e)^  [R-^  Ae-Vx 


stress  in  CC  = 


Fig.  154. 


2   '    e  \2  I 


(7) 
(70 


When  X  =      the  last  term  of  eq.  (7')  is  equal  to  zero  ;  and  for  x  =  o  this  quantity  is 

V  3                                              V  b 
equal      H~  ~  2'  —    '^^ equal  to  The  piece  CC  may  then  be  considered 

as  having  a  uniform  compressive  stress  of  —  and  an  additional  stress  varying  uniformly  from 

zero  at  the  centre  to  a  value  of  —  —  at  each  end,  compressive  on  the  windward  side  and  ten- 
sile on  the  leeward. 

Taking  moments  about  F,  we  find  similarly, 


stress  in  DD'  =  '5_±Zf.  -  (R  P){ 


(8) 


tensile  on  the  windward  and  compressive  on  the  leeward  side.    The  flange  stresses  thus  differ 
R 

numerically  by—-,  as  they  should,  from  2  hor.  comp.  =  o. 

The  shear  on  the  section  EF=  V,  and  may  be  distributed  equally  over  the  web  members 

cut. 

The  stress  in  the  strut  AA'  is  the  same  as  in  the  first  case. 

Form  4.  In  small  bridges,  for  lack  of  head  room,  simple  knee-braces,  DE  and  D'E'  (Fig. 

155),  are  often  used.  p  L^j  ^ 

With  the  same  notation  as  before,  we  have  the  following  stresses:  ^'^^f  e\ 

Passing  the  section  lui  through  (7  and  D,  we  have,  by  taking  moments   d  O 

of  the  forces  on  the  right,  about  C, 

HX  c     {R  +  P)c 


hor.  comp.  tensile  stress  in  ED 

^  e  2e 

—  hor.  comp.  compressive  stress  in  E'D'. 

R4-P 

Bending  moment  at  D  and  D'  —  H  X(c  ~  e)—  {c  —  e). 


(9)  J 

H' 


-b  


/  H 

Fig.  155 


1 


■  ■  (10) 


From  2  vert.  comp.  =  o  on  the  right  of  the  section  pg,  we  have, 
direct  compression  in  CD  —  vert.  comp.  ED  —  V, 


The  direct  compression  in  A'D'  =  V"  =  {R-\-  P)^, 


;i2) 


LATERAL  TRUSS  SYSTEMS. 


"3 


a  value  greater  or  less  than  that  given  by  (i  i)  according  as  f  is  greater  or  less  than  — .  The 

greater  of  the  two  values,  (ii)  or  (12),  is  to  be  added  to  the  compression  due  to  dead  and 
live  loads,  and  the  resulting  fibre  stress  added  to  that  due  to  the  bending  moment  given  in 
eq.  (10).    This  will  give  the  maximum  stress  in  the  end  posts. 

Bending  moment     E  —  H  X  c  —  Fx  f  =  c{R      P)Q  —-^  (13) 


Direct  compression  in  EE'  =  R  -\-  —  —  H 


R 
2 ' 


Direct  compression  in  EC  =  R        -\-  hor.  comp.  ED  —  H  =  — -)-^— -. 


Direct  tension  in  E'C  = 


R-\-Pc 


R 
2' 


(14) 

'IS) 
(16) 


2  e 

The  maximum  compressive  stress  in  CC  therefore  occurs  just  to  the  right  of  E  where 
the  stress  due  to  (15)  is  added  to  that  due  to  the  bending  moment  of  eq.  (13).  Likewise  eqs. 
(13)  and  (16)  give  the  maximum  tensile  stress  in  CC,  which  is  just  to  the  left  of  E'. 

Form  5.  The  bending  moments  on  the  posts  and  the  stress  in  the  strut  CC  may  be 
reduced  without  reducing  head  room  by  the  arrangement  of  Fig. 
156.    The  maximum  bending  moment  in  the  posts  =  H  X  {c  —  e) 
as  before. 

The  stress  at  any  point  F  of  CC  is  given  by  eq.  (7'),  in  which 
e  is  now  the  variable  depth  EF  o[  the  beam.  This  stress  is  a  maxi- 
mum at  the  section  FE  where  the  tangent  at  E  cuts  CC  in  the 
centre  O ;  for  at  that  section  the  variable  lever-arm  e  varies  at  the 

same  rate  as  the  variable  lever-arm      ~  hence  the  rate  of 

change  of  their  ratio  is  zero. 

The  horizontal  component  of  the  stress  in  DD'  is  equal  to  the 

1  his  stress 


H' 


i 


stress  in  CC  at  the  same  section,  minus  —,  as  in  cq.  (8) 


Fig.  156. 


CR+- 


is  zero  at  the  centre,  tensile  on  the  right  and  compressive  on  the  left.  The  maximum  value 
of  the  horizontal  component  of  the  stress  in  DD'  is  at  E,  but  the  maximum  value  of  the  stress 
itself  occurs  at  some  point  to  the  right  of  E.  It  may  be  taken  as  that  point  near  D,  to  the 
right  of  which  the  flange  stress  can  be  assumed  as  transmitted  to  the  post  in  a  direct  line  by 
means  of  the  connecting  plate. 

The  web  stresses  in  the  central  portion  are  found  by  dividing  the  shear,  V  —  vert.  comp. 
DD',  by  the  number  of  pieces  cut.    The  two  web  members  meeting  at  any  point  of  the  flange 
should  also  be  able  to  take  up  the  difference  in  the  flange  stress  in  the  two 
adjacent  panels. 

Form  6.  Fig.  157  shows  a  modification  of  Form  5.  The  maximum 
moments  in  the  posts  occur  at  F  and  F' .  The  maximum  stresses  in  CC 
and  DD'  occur  at  G  and  E,  and  are  found  as  before.  The  stress  in  EF 
may  be  found  by  assuming  that  the  stress  at  D  in  DD'  is  equal  to  that  at 
E,  and  that  the  piece  EF  takes  the  remaining  moment.  Thus  with  centre 
of  moments  at  C, 

Fig.  157.  hor.  comp.  EF  X  (/+  e)  =  H  X  c  —  stress  in  ED  X  e. 

The  web  members  between  E  and  E'  take  the  shear  V,  and  to  the  right  of  E  they  take  the 
shear  F,  minus  the  vert  comp.  of  the  stress  in  EF.    At  jSthey  must  take  the  vert.  comp.  in  EF. 
Portals  tvith  Fixed  End-posts. — In  the  foregoing  discussion  the  end-posts  have  been  treated 


114 


MODERN  FRAMED  STRUCTURES. 


as  not  capable  of  resisting  bending  moment  at  their  bases,  A  and  A'  in  the  figures.  Where 
they  are  so  well  anchored  to  the  masonry,  however,  that  the  ends  are  fixed  in  position,  the 
stresses  in  the  portal  and  in  the  posts  are  very  much  reduced.  With  a  well-designed  portal 
the  upper  ends  of  the  posts  are  also  fixed,  and  we  may  take  the  point  midway  between  the 
lower  edge  of  the  portal  and  the  foot  of  the  post  as  the  point  of  inflexion.  Then  in  Fig.  154, 
for  example,  the  reactions  H  and  V  may  be  considered  as  applied  at  a  point  half-way  between 
A  and  D,  at  which  point  the  moment  is  zero.  The  analysis  then  is  the  same  as  already  given, 
the  distance  c  being  replaced  by  -f-  \{c  —  e).  The  bendii)g  moments  on  the  posts  are  thus 
reduced  one-half  and  the  other  stresses  correspondingly.* 

The  requisite  strength  of  anchorage  to  give  fixed  ends  is  readily  found  from  the  bending 
moments  at  the  feet  of  the  posts.    These  are  the  same  as  at  the  points  D  and  D' . 

The  preceding  applies  also  to  the  sway  bracing  of  elevated  structures  where  the  support- 
ing columns  are  firmly  anchored  to  the  foundations. 

116.  Skew  Portals. — Fig.  158  is  a  plan  of  the  upper  lateral  system  of  a  skew  bridge 
with  vertical  end  posts,  and  Fig.  158  (a)  is  the  elevation  of  the  portal  upon  a  parallel  plane. 


(a)    <53  ^ 


A  (a) 


Fig.  158. 

P  P 
The  lateral  force,     +  -  .  applied  at  C  {R  is  the  lateral  component  in  CD,  and  -  is  the  wmd 

pressure  applied  directly  to  C),  is  resolved   into  the  components       ■\-  ^  sec  /?  and 

[r  +  ^)  tan      where  ft  =  angle  of  skew.    The  force  (^R  +       sec  /3  replaces  the  force 

P  /  P\ 

R  -\-  ~  in  the  discussion  of  the  preceding  article,  and  \R      -j  tan  ft,  together  with  a  similar 

force  at  the  right  end,  act  upon  the  main  truss,  producing  only  slight  additional  stresses. 
The  plan  of  the  upper  laterals  and  inclined  portal  of  a  skew  bridge  are  shown  in  Fig.  159. 
The  lower  laterals  are  represented  by  dotted  lines.  Fig.  I59(«)  is  the  projection  of  the  portal 
on  a  plane  parallel  to  itself,  and  Fig.  {/>)  is  the  elevation  of  one  truss.    The  lateral  force  at 

C  in  the  plane  of  the  portal  is,  as  before,  {r -\-  -]  sec  ft.    The  perpendicular  distance  from 

A  A'  to  CC  is  equal  to  V  A^  -\-J^,  where  AE  is  the  horizontal  distance  between  these  two 
lines,  equal  to  AF  cos  ft,  and  h  is  the  height  of  the  truss.  This  distance  corresponds  to  the 
distance  c  of  the  preceding  article. 

117.  Sway  Bracing  of  the  same  form  as  portal  bracing  is  usually  placed  at  each  panel 
point  of  a  deck  bridge  and  at  each  panel  point  of  a  through  bridge  when  the  height  is  more 
than  about  25  feet.  This  bracing  is  sometimes  designed  to  carry  the  wind  pressure  from  one 
chord  to  the  other  at  each  panel  point,  in  which  case  but  one  lateral  system  is  needed. 
The  stresses  are  computed  the  same  as  for  portal  bracing,  the  external  lateral  force  being 
equal  to  the  wind  load  upoa  one  panel.    The  resulting  vertical  reactions  corresponding 

*  For  the  case  of  end  posts  fixed  at  the  base  with  a  simple  system  of  diagonal  bracing  at  top,  see  Art.  151,  Chap.  X. 


LATERAL   TRUSS  SYSTEMS. 


to  Fand  V ,  Art.  115,  act  as  loads,  upward  or  downward,  upon  the  main  trusses.  The  portal 
bracing  proper  is  now  subjected  to  the  same  loads  as  the  intermediate  sway  bracing. 

When  both  lateral  systems  are  used,  the  stresses  in  the  sway  bracing  are  still  often  com- 
puted on  the  same  assumptions  as  the  above,  although  if  the  two  lateral  systems  have  equal 
lateral  deflections  these  stresses  are  zero.  However,  with  wind  pressure  upon  the  unloaded 
bridge,  the  lateral  system  of  the  chord  supporting  the  floor  has,  in  the  case  of  railway  bridges, 
only  about  one  third  of  its  full  load,  while  the  other  system  is  fully  loaded.  In  this  case  the 
lateral  deflections  are  not  equal ;  the  sway  bracing  will  be  distorted,  and  some  stress  will  be 
thereby  transmitted  to  the  stiffer  lateral  system.  The  assumption  that  one  half  the  wind  press- 
ure upon  the  one  system  is  thus  transferred  is  certainly  on  the  safe  side  when  the  portals  are 
properly  designed,  even  with  the  most  rigid  form  of  sway  bracing ;  with  a  flexible  form,  as 
simple  knee-braces,  very  little  stress^  can  come  upon  this  bracing  from  wind  pressure.  No 
change  need  be  made  in  the  laterals  on  account  of  stresses  arising  from  the  sway  bracing. 

118.  Eccentric  Loads. — For  double-track  deck  railway  bridges  and  for  highway 
bridges,  specifications  sometimes  require  that  the  sway  bracing  shall  be  proportioned  to 
distribute  eccentric  loads  equally  upon  the  two  trusses.  Such  a  distribution  it  is  impossible 
to  attain  to  any  extent,  without  the  use  of  extremely  heavy  laterals  and  sway  bracing.  The 
stresses  in  the  three  systems  of  trusses — the  vertical  main  trusses,  the  lateral  horizontal 
trusses,  and  the  sway  bracing — due  to  eccentric  loading,  depend  upon  the  relative  rigidity 
of  the  several  systems,  and  will  now  be  discussed. 

1st.  The  Lateral  Horizontal  Trusses. — The  sway  bracing  of  a  deck  railway  bridge  is  shown 
in  Fig.  160.    Let  /"  be  a  panel  load  upon  one  track,  e  the  eccentricity,  b  the        Fig.  160. 
width  between  trusses,  and  h  the  height  of  the  trusses.    The  portions  of  the     c'  c 

6  e 

load  transferred  to  C  and  C  are  \P-\-  Pj  and  ^P  —  P-^,  respectively,  there 

2e 

being  an  excess  of     ^  at  C  over  that  at  C.    So  far  as  the  stresses  due  to 

2e 

eccentricity  are  concerned  we  may  then  consider  a  load  equal  \.o  P-^  2X  C 
and  no  load  at  C. 

If  there  were  no  horizontal  trusses  in  the  planes  A  A'  and  CC  Fig.  161, 

2e 

the  effect  of  the  load  P-^  would  be  to  deflect  the  truss  A'C  the  full  amount, 

A,  and  to  twist  the  cross-section  into  the  position  shown  by  the  dotted  lines ;  for  the  sway 
bracing  would  still  preserve  the  rectangular  shape  of  the  cross-section,  there  being  no  resist- 
ance to  the  lateral  motion  of  CC  and  A  A'  beyond  the  slight  resistance  of  the  main  trusses 
to  becoming  warped  surfaces.  The  posts  AC  and  A'C  would  thus  be  inclined  as  much  to 
the  vertical  as  CC  and  A  A'  are  to  the  horizontal,  and  if  we  assume  CC  and  A  A'  to  have 

equal  lateral  movements  the  movement  of  each  would  be  ^     j-    If,  however,  there  were 

horizontal  trusses  in  the  planes  AA'  and  CC,  this  lateral  movement  would  be  somewhat 
resisted,  and  with  extreme  conditions  of  perfectly  rigid  sway  bracing  and  very  flexible 

horizontal  bracing  the  lateral  deflection  of  each  horizontal  truss  will  be  less  than  -  A- 

2  b' 

Hence  the  greatest  stresses  that  can  occur  in  the  members  of  the  lateral  horizontal  trusses 

are  less  than  those  due  to  a  deflection  of  ^  ^  ^-    The  stress  per  square  inch  due  to  this 

deflection  is  independent  of  the  size  of  the  members,  but  with  heavy  bracing  the  deflection, 
and  therefore  the  unit  stress,  will  be  reduced.  It  will  be  shown,  however,  that  the  stresses 
due  to  this  maximum  deflection  are  very  small. 


A* 

Fig.  161. 


MODERN  FRAMED  STRUCTURES. 


If  J„  =  deflection  of  the  vertical  truss  under  full  live  load  P, 
=  deflection  of  lateral  truss, 

2  b' 

5„  =  average  unit  stress  in  vertical  truss  for  deflection  J„ 
=  unit  stress  due  to  full  live  load, 

Sk.  —  average  unit  stress  in  lateral  horizontal  truss  for  deflection  A^,  we  have 


Now  for  any  given  case  can  be  computed  by  eq.  (6),  Chap.  XV,  using  for  this  purpose  a 
single  average  value  of  S^,  or  unit  stress,  to  replace pt  and  p,  of  that  equation.  Having  found 
A^,  the  value  of  A^  may  be  found  by  means  of  the  above  equation.  The  corresponding  value 
of  Sh  is  then  determined  by  putting  p^  —  p^  =  S,^  in  the  second  of  eqs.  (7),  Chap.  XV  (assuming 
all  the  deflection  to  be  due  to  the  web  members),  and  solving  for  S/,,  the  deflection  being  known. 

In  ordinary  cases  5„  is  about  8000  lbs.  per  square  inch  ;  and  the  greatest  stress  on  the 
laterals  due  to  eccentric  loading  is  not  usually  greater  than  2000  lbs.  per  square  inch,  an  insig- 
nificant amount  as  compared  to  the  great  unit  stress  allowable  in  these  members.  In  through 
bridges  of  average  depth  the  ordinary  form  of  sway  bracing  is  very  flexible,  and  the  stress  in 
the  laterals  is  much  less  than  the  above. 

Take  as  an  example  a  Pratt  truss  of  200  ft.  span  ;  12  panels ;  ^  =  33  ft.  4  in. ;  b  =  28  ft. ; 
^  =  6  ft.  The  actual  computed  deflection  for  full  live  and  dead  loads  is  about  2.5  in.  If  one 
half  the  dead  load  (double  track)  be  taken  at  1300  lbs.  per  foot,  and  the  live  load  per  track  at 
3000  lbs.  per  foot,  the  deflection  due  to  full  live  load  is  equal  to 

3000 

A^  =          X  2.5  =  1.8  m. 

4300  ^ 

and 

e/i  ^  6X33-^ 

Now  the  actual  deflection  of  the  lateral  system  in  terms  of  the  unit  stress,  computed  as 
in  Chap.  XV  (assuming  all  the  deflection  to  be  due  to  the  web  members  because  of  the  great 
size  of  the  chords),  is  equal  to 

A;,  =  .0002  5  S,. 

Hence 

Si,  —  4000 J/,  =  1 840  lbs.  per  square  inch. 

2d.  The  Sway  Bracing — It  has  been  shown  above  that  with  no  lateral  bracing  there  is 
no  appreciable  stress  upon  the  sway  bracing;  also  it  is  clear  that  whatever  stress  is  brought 
upon  the  sway  bracing  must  be  resisted  by  the  laterals  at  C  and  A.  But  the  greatest  resist- 
ance that  is  offered  by  these  laterals  is  that  due  to  a  unit  stress  of  possibly  2000  lbs.  per  square 
inch,  an  amount  equal  to  about  one-seventh  the  stress  due  to  wind  pressure.  Hence  the 
stress  in  the  sway  bracing  is  not  more  than  would  be  caused  by  the  transferring  of  one 
seventh  the  wind  pressure  at  each  panel  from  one  chord  to  the  other,  an  insignificant  amount. 

Finally  it  may  be  said  that,  besides  transferring  perhaps  one-half  the  wind  pressure  as 
shown  on  p.  115,  the  ofiice  performed  by  the  sway  bracing  is  to  prevent  independent  lateral 
vibration  and  swaying  of  the  vertical  trusses;  also  to  stiffen  long  posts  and  to  aid  in  the 
erection.     Its  desig'n  must  be  left  mainly  to  the  judgment  of  the  engineer. 


LATERAL   TRUSS  SYSTEMS. 


In  deck  bridges  and  deep  through  bridges  the  rectangular  shape  of  the  cross-section  will 
be  almost  exactly  preserved  by  the  sway  bracing,  but  with  shallow  sway  bracing  or  with  none 
at  all,  the  floor-beams,  when  rigidly  connected  to  the  posts,  are  called  upon  to  act  as  sway 
bracing,  and  small  bending  moments  are  produced  in  them  with  some  tension  on  the  connect- 
ing rivets.    The  stresses  thus  produced  are  very  small,  as  shown  above. 

119.  The  Centrifugal  Force*  F,  of  a  body  of  weight  P,  moving  in  a  curve  of  radius  r, 
is  equal  to 

v'' 

F=-P,  (17) 

where^is  the  acceleration  of  gravity  =  32.2  ft.  per  second,  =  32.2  X        ^q^^  =  79,ioo  miles 

5  280 

per  hour.  Expressed  in  miles,  r  =  ^  5280,  where  5730  =  radius  in  feet  of  a  1°  curve, 
and  D  =  degree  of  curve.  Hence 

F  =  X  5280^^  ^  .00001 1 7Z/'/?/'  =  kP,  (18) 

5730  X  79100 

where  v  is  in  miles  per  hour  and  D  —  degree  of  curve  ;  P  and  F  are  in  like  units. 

Fig.  162  represents  a  transverse  section  of  a  through  bridge.  Tiie  centre  of  gravity  of 
the  panel  load  is  at  G,  with  an  eccentricity  e.  The  eccentricity  e  is  the  average  c'  c 
eccentricity  for  one  panel  length  and  is  due  partly  to  the  eccentricity  of  the 
track  and  partly  to  the  horizontal  displacement  of  the  centre  of  gravity,  caused 
by  the  inclination  of  the  track.  It  will  be  taken  as  positive  outwards.  The  line 
DD'  is  the  centre  of  the  floor-beam;  AA'  and  are  lateral  struts.  The 
stresses  caused  by  the  force  /^and  the  load  P,  in  the  laterals,  the  main  trussses, 
the  stringers  and  floor-beams,  will  be  discussed  in  order.  ^.  ^ 

Tlic  Lateral  Trusses. — The  loads  F  and  P  are  given  over  to  the  floor-beam      Fig.  162. 
and  thence  to  the  posts  at  D  and  D' .    Fig.  163  shows  the  floor-beam  with  reactions.  The 

F 

-iXoG  horizontal  reactions  are  each  equal  to  -,  assuming  them  equal  to  each 

1  p'\    n       Dp   other.    These  horizontal  forces  are  carried  by  the  posts  to  the  laterals, 

2  v'!^  h  *lv  ^  h~a 

Fig.  163.  the  portion  taken  by  the  lower  laterals  being  equal  to X  — ^ — ,  and  that 

taken  by  the  upper  being  equal  to     X    >  one-half  of  each  being  applied  at  each  side.  The 

bending  moments  at  D  and  D'  =  -   ^. 

2  /i 

If  the  live  load  is  taken  as  a  uniform  load,  then  P  is  the  same  for  all  panels,  and  likewise 
F.  But  if  wheel  loads  are  used,  then  Pand  F  vary.  However,  since  F  is  a  constant  function 
of  P  for  any  one  problem,  the  maximum  moments  and  shears  in  the  lateral  systems,  due  to 
centrifugal  force,  will  be  a  constant  function  of  the  maximum  moments  and  shears  in  a  vertical 
truss  due  to  the  actual  wheel  loads.    Hence,  for  the  lower  laterals,  to  get  moments  and  shears, 

multiply  those  found  in  a  vertical  truss  for  the  total  wheel  loads  by  ;  and  for  the  upper 

laterals  multiply  by       .    The  maximum  value  of  F  for  one  panel  is  equal  to  the  maximum 

vertical  floor-beam  load  multiplied  by  k.  This  value  of  F  is  to  be  used  in  getting  the 
moments  on  the  posts. 

Usually  one  of  the  lateral  systems  lies  practically  in  the  plane  of  the  floor-beams,  in  which 
case  the  whole  of  F  is  carried  by  this  one  lateral  system  and  the  posts  receive  no  bending  moment 

*  For  a  fuller  discussion  of  this  subject  see  a  paper  by  Prof.  Ward  Baldwin,  M.  Am.  See  C  E    Trans  Am  See 
C.  E.,  Vol.  XXV,  p.  459. 


11^ 


MODERN  FRAMED  STRUCTURES. 


The  vertical  main  tr?isses  supply  the  reactions  V  and  V,  Fig.  163.  Taking  moments 
about  D',  we  have 

K="l:;h^^=.(i_i±i/).  

and  from  2  vert.  comp.  =  o, 

^'  =  ^^  +  ^'0   .    .    .  (.0) 

From  these  two  equations  we  see  that  the  inner  truss  receives  its  maximum  load  for  a 
minimum  value  of  k,  that  is,  for  a  stationary  load  ;  and  that  the  outer  truss  receives  its  maxi- 
mum load  for  a  maximum  value  of  k. 

Since  e  varies  for  the  difTerent  panels,  Fand  V  are  not  constant  functions  of  P.  How- 
ever, those  portions  of  Fand  V  not  containing i.e.,        —        and  ^(j  + )>  constant 

functions  of  P,  and  the  stresses  in  the  trusses  due  to  these  portions  of  the  load  may  be  found 
■  similarly  to  the  stresses  in  the  laterals;  that  is,  by  finding  the  moments  and  shears  in  a  single 
truss,  or  in  this  case  the  stresses,  due  to  the  actual  total  wheel  loads,  and  then  multiplying  by 

the  factor  ^  for  the  inner  truss  {k  =■  o),  and  by      -\-        for  the  outer  truss. 


For  the  stresses  due  to  the  portions  —  P-^  and  -|-  P-^  it  is  sufficiently  accurate  to  assume  an 

equivalent  uniform  load,  determine  e,  and  thence  P^  for  each  panel,  and  with  these  panel  loads 

compute  the  stresses  in  the  ordinary  way.   A  portion  of  the  loads  on  each  truss  will  be  upward. 

Stringers. — Each  stringer  will  receive  a  lateral  moment  and  shear  equal  to  k  times  the 
maximum  vertical  moment  and  shear  due  to  one-half  the  actual  wheel  loads.  This  moment 
is  taken  by  the  upper  flange  and  adds  to  its  stress,  and  the  shear  requires  additional  rivets  to 
connect  the  stringer  to  the  floor-beam. 

The  vertical  load  upon  each  stringer  is  found  just  as  for  the  trusses,  and  is  given  by  eqs. 
(19)  and  (20)  by  substituting  for  b  the  distance  between  stringers.  If  we  call  this  distance  b' , 
the  load  upon  the  inner  stringer  is,  for  k  —  o, 

PI  2e\ 


7      ^  +  ¥\ 


and  that  upon  the  outer  stringer  is 

2\    '  b' 

In  this  case  e  may  be  taken  as  constant  for  one  panel,  using  an  average  value.  The  maxi- 
mum moments  and  shears  in  the  stringers  are  then  found  by  multiplying  those  for  actual 

.    .  P. 

wheel  loads  symmetrically  placed,  by  the  multipliers  of  —  in  the  equations  above.    The  quan= 

tity  e  refers  in  this  case  to  the  axis  of  the  stringers  and  not  necessarily  to  the  axis  of  the 
bridge;  it  will  thus  have  different  values  in  different  panels,  and  hence  the  stresses  in  the 
stringers  will  be  different. 

Floor-beams. — The  total  maximum  floor-beam  load  is  that  due  to  the  actual  wheel  loads 
moving  on  a  straight  track.  The  portions  given  over  to  the  two  trusses  are  given  by  eqs.  (19) 
and  (20).  With  k  —O,  the  value  of  F  is  the  maximum  shear  at  the  inner  end  ;  and  with  k  a 
maximum,  the  corresponding  value  of  V  is  the  maximum  shear  at  the  outer  end  of  the  floor- 
beam.  These  shears,  multiplied  by  the  distances  between  the  posts  and  the  stringers,  give 
the  maximum  moments  in  the  floor-bealns.  The  value  of  e  in  the  above  is  to  be  taken  as  the 
average  value  for  two  panels,  referred  to  the  axis  of  the  bridge. 

The  floor-beams  are  also  subjected  to  a  direct  compression  from  the  laterals. 


FUNDAMENTAL  RELATIONS  IN  THE  THEORY  OE  BEAMS. 


CHAPTER  VIII. 
FUNDAMENTAL  RELATIONS  IN  THE  THEORY  OF  BEAMS. 

120.  Historical  Sketch.* — For  two  hundred  and  fifty  years  the  true  theory  of  the  strength  of  a  beam 
has  been  a  much-mooted  question  amongst  physicists,  engineers,  and  mathematicians. 

Galileo  was  the  first  of  whom  we  have  any  record  who  undertook  to  discuss  the  problem.  In  his 
famous  Dialogues  (Leiden,  1638,  from  which  Fig.  164  is  taken)  he  propounds  a  theory  based  on  an  assumed 
absolute  rigidity  of  the  material,  and  concluded  that  the  fibres  of  the  beam  were  subjected  to  a  uniform 


Fig.  164. 

tension  which  acted  about  the  base  of  the  beam  as  a  fulcrum.    On  this  theory  the  moment  of  resistance  of 

a  solid  rectangular  beam  would  be  where  /is  the  ultimate  strength  of  the  material  in  tension. 

Robert  Hooke  first  published  his  famous  law  of  the  relation  between  strain  and  stress  in  1678, 
discovered  by  him  he  says  18  years  previously,  and  kept  secret  for  the  purpose  of  procuring  patents  on  some 
applications  of  the  principle  to  springs  for  watches,  clocks,  etc.  Two  years  previously  he  had  ventured  to 
publish  the  law  in  an  anagram  at  the  end  of  another  book,  in  this  form,  "  c  it  i no s  s st t  uv,"  which  being 
interpreted  reads,  "  Ut  tensio  sic  vis,"  or,  "as  the  extension  so  is  the  resistance."  Hooke  makes  this  law 
apply  to  all  "springy"  bodies,  amongst  which  he  names  nearly  all  ordinary  solids.  This  is  still  known  as 
Hooke' s  Law. 


*This  historical  review  of  the  development  of  the  true  theory  of  the  beam  is  derived  mostly  from  Saint-Venant's 

"  Historique  Abrdg/  iles  Recherclies  sur  la  Resistance  et  sur  I' Elasticiti!  des  Corps  Solides"  prefixed  to  his  Nacvier's 
"  Lemons,"  Third  Edition,  Paris,  1864,  and  from  Todhunter's  "History  of  the  Theory  of  Elasticity,"  Cambridge,  Eng., 
i886. 


120 


MODERN  FRAMED  STRUCTURES. 


Mariotte  showed  by  experiment  in  1680  that  the  fibres  on  one  side  of  the  beam  were  extended  and  on  the 
other  side  compressed,  and  assumed  that  the  neutral  surface  passes  through  the  centre  of  gravity  of  the 
section. 

Varignon,  in  J 702,  undertakes  to  harmonize  the  theories  of  Galileo  and  Mariotte,  by  admitting  the 
extension  of  the  fibres,  but  puts  the  neutral  plane  at  the  bottom,  as  Galileo  did,  and  assumes  the  tensile  stress 
as  uniformly  varying  from  there  to  the  other  side.    This  would  make  the  strength  of  a  solid  rectangular 

beam   ,  which  agrees  almost  exactly  with  the  facts  for  cast-iron  at  rupture  when  /  is  the  tensile  strength. 

3 

James  Bernouilli  made  an  important  advance  by  applying  Mariotte 's  law  to  obtain  deflections  of  beams 
(1694  and  1705),  and  argued  that  the  position  of  the  neutral  axis  is  a  matter  of  indifference,  which  was  a 
great  error.  He  denied  the  truth  of  Hooke's  law,  which  we  know  is  not  applicable  to  all  substances, 
nor  to  the  point  of  rupture  with  any  substance.  He  first  constructed  stress-strain  curves,  but  his  work  in 
the  field  of  hydraulics  was  of  even  greater  importance  than  in  the  study  of  solids. 

A.  Parent,  a  French  academician,  seems  to  have  been  the  first  to  perceive  (1713)  the  mechanical 
necessity  of  equilibrmm  between  the  tensile  and  compressive  stresses,  which  condition,  together  with  that  of 
a  uniform  variation  of  stress,  fixes  the  position  of  the  neutral  axis  at  the  centre  of  gravity  of  the  section. 
This  important  discovery  seems,  however,  to  have  passed  unnot  ced. 

Coulomb  reannounced  this  relation  in  a  memoir  to  the  French  Academy  in  1773,  or  sixty  years  after  its 
first  publication  by  Parent.  Saint-Venant  credits  Coulomb  with  never  having  seen  Parent's  work,  as  no 
writer  of  that  century  has  mentioned  it.  But  even  after  this  second  publication  of  so  important  a  necessary 
truth,  such  workers  as  Girard,  Barlow,  and  Tredgold  all  misconceived  the  mathematical  necessities  in  the 
problem,  and  resorted  to  various  makeshifts  to  explain  the  strength  of  beams. 

Navier  finally,  in  1824,  put  the  matter  on  a  solid  mathematical  basis,  although  he  also  at  first  went 
entirely  astray.  He  stated  in  his  first  edition  that  the  moment  of  resistance  varied  as  the  cube  of  the  depth 
of  the  beam,  and  in  his  second  edition  this  error  was  corrected,  but  the  moment  of  the  stresses  on  one  side 
the  neutral  axis  was  said  to  be  equal  to  the  moment  of  the  stresses  on  the  other  side,  about  that  axis, 
an  equality  which  does  not  exist  except  on  symmetrical  sections.  Navier  also  fully  developed  the  theory 
of  the  deflection  of  beams  as  we  now  use  it. 

Saint-Venant,  a  student  of  Navier's,  has  finally  (1857)  in  his  notes  on  Navier's  Lemons  given  a  complete 
analysis  of  both  the  elastic  and  the  ultimate  strength  of  a  beam,  with  suitable  equations  which  will  give 
theoretical  results  agreeing  with  the  actual  tests,  when  the  empirical  constants  are  prbperly  evaluated. 
This  great  engineer,  physicist,  and  teacher  has  done  more  than  any  other  one  to  bring  theory  and  practice 
into  harmony  and  to  put  both  on  a  thoroughly  scientific  basis,  so  far  as  the  strength  a.id  elasticity  of 
engineering  materials  is  concerned.* 

In  spite  of  these  various  true  expositions  of  this  subject  the  source  of  strength  in  a  beam  * 
continues  still  to  be  very  imperfectly  understood  by  many  engineers,  and  even  by  current 
writers  on  applied  mechanics,  and  gross  errors  in  this  direction  are  still  common.    It  is  in 
consideration  of  this  state  of  the  science  that  the  problem  is  treated  so  fully  here. 

121.  Elementary  Principles  of  Universal  Application. f — Since  stress  is  the  internal 
resistance  to  distortion  produced  by  the  application  of  external  forces  to  a  body,  there  is 
always  an  equilibrium  established  between  the  internal  stresses  and  the  external  forces. 
When  the  body  is  a  rigid  beam  which  is  acted  upon  by  external  forces  which  produce  shear 
and  bending  moment  at  any  given  section,  the  equilibrium  between  the  stresses  and  forces  is 
of  exactly  the  same  kind  as  obtains  in  the  case  of  a  framed  structure,  as  discussed  in  Chapter 
II.  Hence  the  three  general  equations  of  equilibrium  apply  to  beams  the  same  as  they 
do  to  trusses.  We  may  therefore  pass  a  section  through  a  beam,  replace  one  portion  by  the 
stresses  acting  at  this  section,  and  write  the  three  general  equations : 

'2  vertical  components  of  stresses  —  2  vertical  components  of  external  forces ; 
2  horizontal       "  "       "      =:  2  horizontal       "         "        "  " 

2  moments  "       "      =2  moments  "        "  " 


*He  died  January  6,  1886. 

f  In  this  chapter  few  proofs  are  given  for  fundamental  equations,  these  being  fully  developed  in  modern  text 
books  on  Applied  Mechanics. 


FUNDAMENTAL  RELATIONS  IN  THE  THEORY  OF  BEAMS.  121 


A 

7 


The  sum  of  the  vertical  components  of  the  stresses  is  called  the  shear  on  the  sectioa 
When  the  beam  is  subjected  to  vertical  forces  only,  there  is  no  result- 
ing horizontal  component  either  of  the  forces  or  of  the  stresses.  That 
is  to  say,  the  algebraic  sum  of  the  horizontal  stresses  which  make  up  ^ 
the  resisting  moment  must  be  zero,  since  there  is  no  resultant  hori-  ^ 
zontal  force  to  be  resisted.   Therefore  the  sum  of  tlie  compressive  stresses  |~  b 

must  always  equal  the  sum  of  the  tensile  stresses  in  simple  cross-bending.  105 
This  is  a  mathematical  or  mechanical  necessity,  and  holds  true  at  rup- 
ture the  same  as  at  the  elastic  limit,  being  independent  of  the  material  or  of  the  form  of  the 
section.  If  these  two  opposing  stresses  were  concentrated  at  their  respective  centres  of 
gravity,  they  would  form  a  couple,  the  moment  of  which  is  the  moment  of  resistance  which 
holds  in  equilibrium  the  external  forces,  and  hence  it  is  always  equal  to  the  external  moment. 
The  horizontal  surface  where  the  stresses  change  from  tension  to  compression  is  called  the 
neutral  surface,  or  neutral  axis,  since  here  there  is  neither  tension  nor  compression. 

If  the  centre  of  moments  be  taken  at  this  surface,  the  moment  of  the  resisting  couple  is 
the  arithmetical  sum  of  two  moments  which  are  equal  to  each  other  under  symmetrical  condi- 
tions. But  since  these  partial  moments  are  added  together  to  make  up  the  total  moment  of 
resistance,  there  is  no  logical  necessity  that  they  should  be  equal,  and  when  the  cross-section 
is  not  a  symmetrical  one  they  never  are  equal. 

122.  Elementary  Principles  True  within  the  Elastic  Limit. — Within  the  elastic 
limit  the  ratio  between  stress  and  strain  is  a  constant  one,  or  here  Hooke's  Law  holds  true. 
Beyond  this  limit  the  stress  does  not  increase  as  rapidly  as  the  strain. 

Within  the  elastic  limit  also  a  normal  section  of  the  beam  which  is  plane  before  bending 
is  a  plane  and  nearly  normal  after  bending.  Since  after  bending  two  plane  sections  which 
originally  were  parallel  become  inclined  to  each  other,  it  follows  that  the  fibres,  or  elements, 
joining  these  planes  have  a  uniformly  varying  length  across  the  section  even  though  these 
planes  are  no  longer  normal.  If  the  bending  has  stopped  within  the  elastic  limit,  then  the 
stresses  are  proportional  to  the  strains  they  are  resisting  and  therefore  for  all  bending  zvithin 
the  elastic  limit  the  stresses  are  uniformly  varying  across  the  section.  This  conclusion  is  a 
logical  or  geometrical  necessity  from  the  premises  which  have  been  experimentally  estab- 
lished. 

From  the  two  conditions  of  uniformly  varying  stress  and  the  absolute  equality  between 
the  sum  of  the  tensile  and  the  sum  of  the  compressive  stresses,  it  follows  (from  the  laws  of 
mechanics)  that  the  neutral  axis,  or  neutral  plane,  traverses  the  centre  of  gravity  of  the  section. 

li  I  =  moment  of  inertia  of  the  cross-section, 

/  =  unit  stress  on  the  extreme  fibre  on  either  side, 

y^  =  distance  from  neutral  axis  (c.  of  gr.)  to  the  same  extreme  fibre, 

=  moment  of  resistance  of  beam  at  any  section,  =  bending  moment  M,  of  the 
external  forces  on  one  side  of  that  section  about  a  centre  taken  at  the  neutral 
axis  in  the  plane  of  the  section, 

then,  so  long  as  /remains  inside  the  elastic  limit,  we  have,  for  all  materials  and  forms  of  cross- 
section,  the  following  exact  and  universally  true  relation : 

M=M^=-^^,  ^  (I) 

which  is  the  general  equation  of  the  moment  of  resistance  of  a  beam,  within  the  elastic  limit.^^^;r~^  ^2 

If  the  section  is  a  symmetrical  one,  f  and  y^  are  the  same  for  the  extreme  fibres  on  both 
the  tension  and  the  compressive  sides.  If  unsymmetrical,  as  in  Fig.  166,  then  the  extreme 
fibre  on  the  compressive  side,  being  much  farther  away  from  the  neutral  axis,  has  a  corre- 


122 


MODERN  FRAME  I)  STRUCTURES. 


spondingly  larger  stress.  The  unit  stresses  follow  a  law  of  uniform  variation  across  the  entire 
J-  <        section,  but  total  stress,  being  the  unit  stress  multiplied 


J  L 


1 


unit  stress 


by  the  area  over  which  it  acts,  might  be  shown  graphically 
by  multiplying  the  unit  stresses  by  the  corresponding 
widths  of  the  section,  which  would  give  a  diagram,  as  shown 
Fig  167  ^^S-  ^^7'  where  the  areas  above  and  below  the  neutral 
axis  are  equal. 

The  following  moments  of  resistance  of  solid  cross-sections  in  terms  of  the  unit  stress  on 
the  most  distant  fibre,  with  the  corresponding  values  of  /  and  ,  are  rigidly  correct  for  all 
bending  stresses  inside  the  elastic  limit  of  the  material. 


Fig.  166. 


Figure. 

Distance  of  Centre  of  Gravity, 
or  Neutral  Axis,  from  the 
most  distant  fibre 

=  yi- 

Moment  of  Inertia  about  the 
Centre  of  Gravity 
of  the  Section 
=  /. 

Moment  of  Resistance 
in  terms  of  the  Stress  in  the 
most  distant  fibre 

yi 

168. 

. — h 

1 

--t- 
1 

\. 

h 
2 

12 

Ifbh^ 

169. 

d 
2 

nd'^ 
64 

32-' 

t 

?/ 

170.  -'L 

'     /  \ 

h— 6 — ' 

i 

h 

—  1 
\  * 

bk^ 
36 

171.  -i-<^  

/i 

-  V2 
2 

k* 
12 

J'h^ 

t^2 

t 

172.  i 

Ut 

_J'"T" 

h 

h 
2 

bp  -  (b  -  t')(/t  —  2tf 

bh-'  —  (b-  t')()l  -  2ff 

1—' 

12 

\fh'^  +  t{b  -  t'){/i  -  It) 

t'h^l{h  -  V) 

3 

yi 

173.  ! 

i 

t 
i 

f 

.,4.  i/ 

\^  b  J 

 -\— 

b^2b'  h 

...[sb  +  b'    {b  +  2by-\ 

/^'[3iib-^  b'){b  +  V) 
6  [        2(^5  +  2b') 

-  {b-\-  2b') 

_ 

\  1 

/'  +  b'  3 

y   12     i?,(b  +  b') 

FUNDAMENTAL  RELATIONS  IN  THE  THEORY  OF  BEAMS. 


123.  Elementary  Principles  true  beyond  the  Elastic  Limit  and  at  Rupture. — In  Fig. 
175  are  shown  strain  diagrams  of  some  of  the  more  common  materials  used  in  engineering 
structures.  Their  distortions  within  their  respective  elastic  limits  are  too  small  to  be  shown 
on  this  diagram.  These  limits  are  therefore  at  the  points  where  the  curves  leave  the  vertical 
axis.  Cast-iron  and  timber  have  no  definite  elastic  limits,  their  diagrams  being  curved  from 
the  start.    When  these  materials  are  used  in  beams  of  simple  geometric  form,  as  in  solid  rect- 


10c 

it 

00 

r 

'I 

TV 

rp 

IC 

Al 

90 

m 

'■ft 

1 

i 

c 

D  A  1  M    m  Af^  D 

Al 

MS 

1 

-t 

— 
— 

80 

loo 

( 

1 

V 

-+ 

TO 

Ste 

el 

0 

60 

-i- 

Oi 

u 

0 

-+ 

m 

50 

u 

an 

k 

— 

It 
a 

\ 

1 

c 

40 

1* 

V 

1 

1 

H 

r 

SO 

rw 

r* 

-j 

/ 

—J 

20 

U 

/ 

10 

Od 

C 

DID 

pr 

!SS 

or 

E 

lot 

Q 

_L 

2 

1 

1 

i 

-« 

u 

0 

r 

— 

rir 

lb 

^ 

2( 

0 

3( 

.DC 

0 

» 

4( 

.oq 

10 

■OC 

0 

.oc 

0. 

— 

t- 

in 

s 

-" 

m 

10 

» 

c 

> 

il- 

81 

.a 

10 

^ 

a 

0. 

— -- 

a 

"1 

m 

m 

» 

'J 

]n| 

).d 

# 

A 

10 

A 

10 

1," 

13 

A 

10 

# 

14i 

.(10 

0 

150 

0 

— f 

_. 

.00 

0 

0( 

0 

Fig.  175. 


angular  or  circular  cross-sections,  their  breaking  strength  does  not  conform  to  the  equations 
given  in  the  last  article.  In  the  case  of  the  ductile  metals  like  wrought-iron  and  mild  steel 
such  a  beam  will  bend  cold  through  an  angle  of  180°  perhaps  without  rupture,  and  hence  can- 
not be  said  to  have  any  assignable  "  cros.s-breaking  "  strength.  If  such  a  term  is  used  it  should 
be  named  "  cross-bending  "  .strength,  and  by  this  should  be  understood  its  strength  at  its  elas- 
tic limit.  At  this  point  it  conforms  to  the  laws  of  internal  distribution  of  stress  as  given  in 
the  last  article.  Such  materials  therefore  can  scarcely  be  regarded  as  having  any  definite  or 
assignable  ultima1"e  cross-breaking  strength,  or  modulus  of  rupture. 


124 


MODERN  FRAMED  STRUCTURES. 


m 

\. 

1  >| 

Compression  side  of 

beam 

O 

A 

o 

'  Neutral  ajcis 

a 

>  X 

■  \ 

H 

Tension  side  of 

beam 

^ft  ^ 

O 

Fig.  176. 


With  such  materials  as  timber  or  cast-iron,  however,  the  case  is  different.  Here  the 
beam  does  break  under  definite  loads,  but  these  loads  are  always  much  greater  (perhaps  by  100 
per  cent)  than  the  equations  in  the  previous  article  would  seem  10  indicate.  The  fatal  mistake 
writers  on  mechanics  have  made  has  been  in  ever  using  such  a  form  of  equation  as  (i)  to  ex- 
press the  moment  of  resistance  of  a  beam  at  rupture.  Its  field  of  application  lies  only  within 
the  elastic  limit  of  the  material. 

124.  The  Distribution  of  Longitudinal  Stress  (Tension  and  Compression)  across  the 
Section  of  a  Beam  when  loaded  beyond  its  Elastic  Limit— Since  a  section  plane  before 

bending  remains  sensibly  plane  after  bending,  even 
beyond  its  elastic  limit,  it  follows  that  the  strain  or 
distortion  is  uniformly  varying  across  the  section,  under 
all  circumstances.  From  the  neutral  axis  to  the  ten- 
sion side  of  the  beam,  therefore,  the  stress  increases 
just  as  it  does  in  an  ordinary  strain  diagram,  where 
the  increments  of  strain  are  plotted  to  a  uniform  hori- 
zontal scale.  If  the  beam  fails  in  tension,  the  stress,  at 
rupture,  on  the  extreme  fibre  on  the  tension  side  of  the 
beam  is  the  ultimate  tensile  strength  of  the  material. 
Thus  let  the  curves  OC  and  OD  in  Fig.  176  represent  the  tension  and  compression  strain 
diagrams  of  cast-iron,  for  instance.  The  ordinate  CB  =  ft  is  the  tensile  strength  of  the  iron. 
Since  this  is  the  actual  stress  on  the  extreme  fibre  at  rupture,  we  may  construct  a  vertical  sec- 
tion, or  projection  of  the  beam  on  this  diagram  by  taking  the  ordinate  BC  as  the  tension  side 
of  the  beam,  the  point  O  marking  the  neutral  axis.  Then  OB  is  the  distance  from  the  neutral 
axis  to  the  tension  side  of  the  beam  to  some  scale,  and  for  a  rectangular  cross-section  the  area 
OCB  would  represent  the  total  tensile  stress  in  the  beam  at  that  section.  But  since  the  total 
tensile  must  equal  the  total  compressive  stress  in  simple  bending,  the  area  cut  off  from  the 
indefinite  compression  strain  diagram  OE  by  the  line  marking  the  position  of  the  com- 
pressed side,  AD,  must  be  just  equal  to  the  area  OCB,  or  OAD  —  OCB.  This  condition 
fixes  the  position  of  that  side  of  the  beam,  AD,  to  the  same  scale  as  obtains  for  the  distance 
OB,  and  hence  AB  represents  the  total  height  of  the  beam,  and  the  scale  of  the  drawing 
is  determined.  The  point  O  is  then  the  position  of  the  neutral  axis  at  rupture,  and  the 
ordinates  to  the  lines  OC  and  OD  from  the  axis  AB  represent  the  tensile  and  compressive 
stresses  across  the  section. 

The  moment  of  resistance  is  the  moment  of  the  couple  formed  by  these  equal  and  oppo- 
site stresses,  when  concentrated  at  the  centres  of  gravity  of  their  respective  areas,  which  moment 
of  resistance  should  equal  the  bending  moment  of  the  external  forces  at  rupture.  The  fact 
that  this  moment,  so  computed,  is  not  equal  to  the  breaking  moment  in  the  case  of  cast- 
iron  can  be  attributed  only  to  the  existence  in  cast-iron  of  high 


internal  stresses,  due  to  the  fact  that  the  external  surfaces  are 
solidified  first.  These  outer  fibres  are  all  in  a  state  of  initial 
compression,  which  must  first  be  overcome  on  the  tension  side 
of  a  cast-iron  beam  before  these  fibres  begin  to  resist  any 
tensile  distortion.  In  the  case  of  a  wooden  beam  the  action 
is  the  direct  reverse  of  the  above.  Timber  is  much  stronger  in 
tension  than  in  compression  ;  hence  a  wooden  beam  fails  first  on 
the  compression  side  by  breaking  down  the  fibres,  which  results 
in  a  continual  lowering  of  the  neutral  axis,  as  the  load  increases, 
until  the  beam  finally  ruptures  on  the  tension  side  and  the 
failure  is  complete.  Fig.  177  shows  a  common  method  of  fail-  '77- 
ure  of  green  timber  where  the  tensile  stress  at  rupture  AD  is  about  five  times  the  com- 


] 

D 

s 

« 
B 

Axis 

m 
2 

00 

A 

'  ^! 

a 
0 

C 

0 

Neutn 

£ 

Comi 

FUNDAMENTAL  RELATIONS  IN  THE  THEORY  OF  BEAMS. 


pressive  stress  CB.  In  dry  timber  the  compressive  strength  is  greatly  increased  and  the 
neutral  axis  remains  nearer  the  centre  of  the  beam.  When  the  complete  tensile  and  com- 
pressive strain  diagrams  of  any  material  are  known  the  moment  of  resistance  of  a  beam 
of  that  material  can  be  found  as  here  described. 

125.  Rational  Equations  for  the  Moment  of  Resistance  of  a  Beam  at  Rupture. — 
Since  a  strain  diagram  is  a  smooth  curve,  it  would  seem  possible  to  obtain  some  simple  equa- 
tion for  such  a  locus  and  then  proceed  to  find  the  enclosed  areas  and  the  moment  of  the. 
couple  in  terms  of  the  co-ordinates  of  these  curves.  M.  Saint-Venant,  in  his  notes  on 
Navier's  "  La  Resistance  des  Corps  Solides,''  has  offered  such  equations  which  prove  to  be  capa- 
ble of  expressing  the  strength  of  a  beam  at  rupture  very  closely.  Thus  when  the  strain 
diagram  is  fitted  to  the  section  of  a  beam,  as  shown  in  Fig.  178, 

Let  ft  =  tensile  strength  =  length  of  tension  ordinate  where 
curve  becomes  horizontal ; 
/c  =  same  in  compression  ; 

jfi  =  distance  from  neutral  axis  to  tension  side,  if  failure 
occurs  in  tension  ; 
=  distance  from  neutral  axis  to  the  fibre  which  would 
have  stress  (beyond  the  limits  of  the  beam  if 
failure  occurs  in  tension); 
/—  stress  in  fibre  distant  /  from  neutral  axis  in  tension 
side ; 

/'  =  same  at  distance  j/'  in  compressive  side. 

Now  since  nearly  all  bodies  distort  or  flow  somewhat  under  a 
constant  load  at  the  point  of  rupture,  whether  in  tension  or  com- 
pression, the  strain  diagram  would  be  horizontal  at  this  point. 
Also,  as  soon  as  this  point  is  reached,  failure  is  sure  to  occur  under 
a  constant  loading.  Therefore  the  strain  diagram  representing 
the  distribution  of  the  stress  on  the  side  which  fails  may  be  sup- 
posed to  come  into  a  horizontal  position  at  the  outer  fibre  on 
that  side.  On  the  other  side  it  comes  into  a  horizontal  position 
at  a  point  beyond  the  limits  of  tiie  beam.  Thus  both  of  these  curves  may  be  fairly  repre- 
sented by  parabolas  of  various  degrees,  all  having  the  vertices  at  the  point  of  greatest  stress 
and  passing  through  the  origin. 

Saint-Venant  proposes,  therefore,  the  following  equations  for  these  curves : 

/=4--('-f]      /-/-[■-(■-f]  (^) 

The  first  of  these  equations  is  for  the  tension  and  the  second  for  the  compression  side  of  the 

beam.  The  form  of  thesecurves  for  different  values 
of  m  is  shown  in  Fig.  179,  where  they  are  drawn 
only  for  the  failing  side  of  the  beam.  When 


m  =  I  we  have  f  —  ~  y,  which  is  a  linear  function, 

yr 

and  corresponds  to  the  ordinary  equations  within 
the  elastic  limit.  The  value  of  must  be  selected 
so  as  to  make  the  curve  correspond  as  nearly  as 
possible  to  the  strain  diagram  of  the  material,  and 
may  be  different  for  the  two  kinds  of  stress. 
Whenever  the  strength  of  the  material  is  very  much  greater  in  one  way  than  in 
the  other,  as  in  cast-iron,  where  the  compressive  strength  is  some  five  times  what  it  is  in 


Fig.  178. 


Fig.  179. 


126 


MODERN  FRAMED  STRUCTURES. 


tension,  and  in  timber,  where  the  reverse  holds  true,  the  neutral  axis  shifts  at  rupture  very 
far  towards  the  stronger  side,  as  shown  in  Figs.  177  and  178,  and  the  portion  of  the  strain 
diagram  utilized  on  this  side  becomes  practically  a  straight  line.  In  this  case  the  formula 
expressing  moment  of  resistance  is  very  much  simplified,  becoming 


+  3)  ,  J  2 

;// 1  , — -  +  4  r  1 — 


(3) 


+  3  +  2  V  2(W+  l) 

Here /is  the  ultimate  strength  of  the  material  on  the  side  which  fails  first,  as  tension  in 
the  rase  of  cast-iron  and  compression  in  the  case  of  timber.  For  different  qualities  of 
cast-iron,  for  instance,  the  tension  strain  diagrams  would  point  to  some  one  of  the  various 
curves  shown  in  Fig.  180,  thus  indicating  what  value  of  m  to  use  in  eq.  (3).  It  will  be  seen 
thai  m  increases  as  the  material  becomes  more  ductile,  and  the  neutral  axis  moves  farther 


Fig.  180. 


and  farther  towards  the  compressed  side  of  the  beam  when  the  elastic  limit  in  compression  is 
relatively  high.  When  m  —  i  we  have  —  — which  holds  true  within  the  elastic  limit 
for  all  materials.    The  following  table  f  gives  values  of  the  moment  of  resistance  for  various 


Value  of  ;«. 

Dist.  of  Neutr.  Axis  from 

yt 

the  weaker  side  =  — . 

h 

Ratio  of  Fibre  Stress  on 
Stronger  Side  to  that  on 
Weaker  Side  at  Rupture. 

Moment  of  Resistance  at  Rupture 
in  terms  of  the  weaker 
strength  =  . 

I 

0. 500 

1. 000 

1. 000  X  — - 
0 

2 

•550 

1.633 

1-417  " 

3 

.586 

2. 121 

1.654  " 

4 

.613 

2.530 

1.810  " 

5 

•634 

2.887 

1.922  " 

6 

.655 

3.207 

2.007  " 

7 

i  =  .667 

I  =  3-500 

II  =  2.074  " 

00 

1. 000 

00 

3.000  " 

*  From  Saint-Venant's  Navier,  §  151,  p.  182,  edition  of  1864,  Paris.    The  equation  giving  moment  of  resist- 
ance for  the  general  case,  from  which  this  is  derived,  is  given  on  p.  179,  but  is  too  complicated  to  be  reproduced  here, 
t  Saint-Venant's  Navier,  Paris,  1864,  p.  182. 


FUNDAMENTAL  RELATIONS  IN  THE  THEORY  OF  BEAMS. 


127 


values  of  m,  and  also  the  position  of  the  neutral  axis  and  the  ratio  of  the  fibre  stresses  on  the 
two  sides  of  the  beam  at  the  time  of  rupture,  when  the  stress  varies  uniformly  from  the 
neutral  axis  outward  on  the  stronger  side,  as  in  Fig.  180. 

The  value  of  m  for  cast-iron  varies  from  3  to  7,  depending  on  the  toughness  or  ductility 
of  the  iron, /being  taken  as  the  tensile  strength  of  the  metal.  For  timber  /would  be  taken 
as  the  compressive  strength,  and  m  taken  as  4  or  5. 

When  the  common  formula  J/„        '  solid  rectangular  sections       —  ^/M'',  is  used 

for  the  ultimate  strength  of  a  beam, /becomes  the  "modulus  of  rupture  in  cross-breaking," 
and  its  value  always  lies  intermediate  between  the  two  moduli  in  tension  and  compression. 
In  timber  it  is  nearly  an  arithmetic  mean  of  these  two  moduli,  while  for  cast-iron  it  is  from 
to  2  times  the  tension  modulus. 

For  practical  purposes  it  is  just  as  well  to  use  the  ordinary  equation  M„  =•<—  up  to  rup- 

ture,  remembering  that  /in  this  case  becomes  the  modulus  of  rupture  in  cross-breaking,  and 
its  value  must  be  determined  by  cross-breaking  tests.  The  true  theory  of  the  ultimate 
strength  of  a  beam  has  been  given  here,  in  order  that  the  student  may  not  conclude  that 
theory  is  unable  to  cope  with  this  problem,  as  is  commonly  supposed,  and  because  it  is  not 
given  in  English  and  American  works  on  applied  mechanics. 

126.  To  find  the  Moment  of  Inertia  of  a  Section  composed  of  Rectangles. — The 
moment  of  resistance  of  an  irregular  cross-section  in  terms  of  the  stress  in  the  extreme  fibre 
can  only  be  found  by  first  finding  the  centre  of  gravity  of  the  section  which  is  always 
traversed  by  the  neutral  axis  until  after  the  elastic  limit  is  passed,  and  the  moment  of  inertia 
of  the  section  about  this  neutral  axis.  When  the  section  can  be 
supposed  to  be  made  up  of  a  series  of  rectangles  the  centre  of 
gravity  and  moment  of  inertia  can  best  be  found  as  follows:  — 

In  Fig.  181,  the  moment  of  inertia  of  the  rectangle  about 
its  own  centre  of  gravity  axis  00'  is  l„  =  j\  bli^  =  -^b{Y  —  yf. 
Its  moment  of  inertia  about  any  other  axis,  as  aa'  parallel  to 
Oa,  is 


T 


li 


Fig.  i8t. 


(4) 

ay? 


Also  the  statical  moment  of  the  area  about  the  axis  aa'  is 


(5) 


k 


T 


b  i — Th  2  • 
bi  H  1^1  I 


Fig.  182. 

follows,  where  Y  represents  the  larger  of  two  successive  values  of  f. 


In  a  compound  form,  made  up  of  rectangles,  as  in 
Fig.  182,  the  total  area  is  the  sum  of  the  areas  of  the 
several  rectangles;  the  total  statical  moment  about  aa'  is 
the  sum  of  the  partial  moments ;  and  the  total  moment 
of  inertia  about  aa'  is  the  sum  of  the  partial  moments  of 
inertia  about  this  axis,  or  we  may  write 

The  most  convenient  method  of  computing  these 
values  is  by  arranging  the  computation  in  tabular  form  as 


128  MODERN  FRAMED  STRUCTURES. 

TABULAR  COMPUTATION  OF  MOMENT  OF  INERTIA  FOR  RECTANGULAR  FORMS. 


b 

h 

y 

A 

2 



y.__y. 

^y3  _  ^«) 

3 

=  1 

o 

o 

/  ^ 

b,k. 

hx' 

—  «1 
2 

3^' 

hx'' 

b^hi 

{hx  +       -  hx'' 

&C. 

(A,  +  h.,Y  -  h,^ 

&c. 

k\  -|-  k'i 

&c. 

&c. 

&C. 

&c. 

&c. 

&C. 

<^r 
oc^  • 

&c. 

&c. 

b. 

&c. 

&C. 

&C. 

&c. 

&c. 

&c. 

b. 

&c. 

&c. 

&c. 

&c. 

&c. 

&c. 

&c. 

&c. 

'2  =  A 


2=  Ma 


2  =  la. 

Also,  the  moment  of 


We  now  have  for  the  distance  to  centre  of  gravity  axis,  D  = 

inertia  about  the  neutral  axis  =  =  —  AD'  =  —  M^D,  where  is  the  gravity  moment 
of  the  area  about  the  axis  a.       r^/'  a/^o-'^er  ///^  M  c/^cyraAi^y^cA.  Par  /cT^ 

The  moment  of  resistance  of  the  cross-section  is 


fL 

> 


where  /  is  the  stress  on  the  extreme  fibre  which  is  at  a  distance  j/,  from  the  neutral  axis,  either 
side  being  taken. 

127.  To  find  the  Centre  of  Gravity  and  the  Moment  of  Inertia  of  any  Irregular 
Section. 

The  following  graphical  method  consists  essentially  in  the  measurement,  in  most  cases, 
of  a  single  area  easily  constructed,  and  with  the  aid  of  a  planimeter  is  very  rapid  and  accurate. 
The  problem  will  be  treated  in  two  parts  : 

1st.  To  find  the  moment  of  inertia  about  any  given  axis. 

2d.  To  find  the  gravity  axis  and  the  moment  of  inertia  about  that  axis. 

1st.  Let  it  be  required,  for  example,  to  determine  the  moment  of  inertia  of  the  rail 
section,  shown  in  Fig.  183,  about  any  given  axis,  as  AA'.  The  actual  operation  would  be  as 
follows : 

For  the  upper  portion  of  the  figure  draw  any  line  OB,  perpendicular  to  AA',  and.  at  some 
whole  number  of  units  k,  from  AA',  draw  the  parallel  BC.  Draw  also  any  number  of  lines 
through  the  given  area  parallel  to  AA',  a.slp,  kb,jc,  etc.,  spacing  them  closer  where  the  outline 
is  irregular  than  where  regular,  and  lay  off  Bi'  =  Oi,  B2'  =  O2,  B$'  =  O^,  etc.  Then  for 
any  point  d,  draw  d^'  intersecting  BC  in  in  ;  then  Otn,  thus  determining  a  point  d"  on  /^d. 
In  like  manner  find  points  e" ,  f" ,  etc.,  corresponding  to  e,  f,  etc.,  and  join  the  new  system  of 
points  by  a  smooth  curve.  The  oblique  construction  lines  need  not  of  course  be  actually 
drawn,  the  required  intersections  only  being  marked.  The  new  curve  for  the  lower  portion 
is  likewise  constructed,  using  a  line  B' C,  at  a  distance  k,  below  AA'  'm  place  of  BC. 

Then  if  A"  represent  the  total  area  of  our  new  figure  above  and  below,  and  the 
required  moment  of  inertia,  we  shall  have 

L  =  k'A". 

Demonstration, — The  general  expression  for  moment  of  inertia  about  an. axis  A  A'  is 


I,=  fbdyf. 


FUNDAMENTAL  RELATIONS  IN   THE  THEORY  OF  BEAMS. 


129 


where  b  —  length  of  strip  parallel  to  AA\  dy  its  width,  and  y  its  distance  from  the  axis.  If  k 
is  a  constant,  we  may  write 

h  =  k  f{bj)dyy.   (I) 


Fio.  183. 


Fig.  184. 


Referring  now  to  Fig.  184,  let  md'  be  drawn  parallel  to  OB  and  then  Op  through  d'. 
Then,  since  6)4  =  Bd^  and  Bm=/^d'  —  b',  we  have  Bp=4d=b.    Therefore,  by  similar  triangles, 


b      k  b'  k 

-7,  =■  ,  and  rry  ■=.— , 
b     y  b  y 


(3) 


V 

Multiplying,  we  get  b"  —  b''.^.    The  above  construction  being  carried  out  as  in  Fig.  183, 

n. 


for  each  horizontal  strip,  we  have,  by  substitutir.g  in  (2), 

/,  =    J  b"dy 


13° 


MODERN  FRAMED  STRUCTURES. 


If  OB  can  be  made  a  line  of  symmetry,  it  will  be  necessary  to  construct  but  one-half  of 
the  new  curve ;  also,  k  should  be  made  of  such  a  length  as  to  give  a  fair-sized  area  to  measure 
and  at  the  same  time  good  intersections. 

2d.  This  problem  can  usually  be  reduced  to  the  first  by  determining  the  centre  of  gravity 
from  considerations  of  symmetry,  or  by  cutting  out  the  section  from  thick  paper  and  balanc- 
ing :\\  ii  knife-edge.  Where  this  cannot  readily  be  done,  as  in  the  case  of  disconnected  parts, 
we  may  proceed  thus  : 

Assume  an  axis  AA\  Fig.  185,  parallel  to  the  unknown  gravity  axis,  and  construct  the 
area  A"  as  before ;  also  at  the  same  time  project  the  points  in,  n,  p,  etc.,  upon  their  corre- 
sponding horizontal  lines,  thus  fixing  points  c' ,  d',  e',  etc.    Join  these  also  by  a  smooth 

curve,  and  let  A^'  be  that  part  of  the  area  of 
this  new  figure  above  the  axis,  and  A^'  the 
lower  portion.  Only  the  upper  right-hand 
portion  is  shown  in  Fig.  185,  but  the  given 
area  may  be  of  any  shape,  and  the  line  OB 
drawn  anywhere.  As  before,  we  shall  have 
/,  =  k'A".  If  now  A  represent  the  total 
original  area,  we  have,  by  taking  moments 
about  AA', 

Distance  of  centre  of  gravity  above  AA'  =  d, 

'  bdy  y 
~A~ 


=/ 


bV\dy 


A 

'J 


(4) 


Fig.  185. 


d  =  k 


From  (3),  b'  =  b^,  and  considering  y 

negative  below  the  ax\s,J^  d'  dy  =  A/  —  A,'. 
Substituting  in  (4),  we  then  have 
A/  —  A,' 


(5) 


being  the  required  moment  of  inertia  about  a  gravity  axis,  we  have 

=  d'A 
(A/  -  A, 


the 


in  which  the  areas  to  be  measured  are.  A,  the  given  area,  and  A'  {A,'  and  A/)  and  A 
two  "  constructed  "  areas. 

If  desired,  the  area  A'  may  be  first  constructed  and  the  centre  of  gravity  located  by  eq.  (5) 
then  a  new  axis  taken  through  it  and  /„  determined  as  in  the  first  case.  This  construction 
then  gives  us  a  method  for  finding  the  centre  of  gravity  of  a  figure  without  considering  the 
other  part  of  the  problem. 

Considered  as  a  section  through  a  loaded  beam,  it  is  interesting  to  note  that  if  we  make 
k  equal  to  the  distance  to  one  extreme  fibre,  taking  a  gravity  axis,  our  area  A'  will  be  such 


FUNDAMENTAL  RELATIONS  IN  THE  THEORY  OF  BEAMS. 


that  if  a  uniform  stress  of  the  same  intensity  as  that  upon  this  outer  fibre  were  appHed,  the 
resulting  moment  would  be  the  same  as  that  upon  the  section.  This  relation  is  fully  discussed 
and  used  to  some  advantage  in  finding  moments  of  inertia  in  Sir  Benjamin  Baker's  "  Strength 
of  Beams,"  the  above  method  being  in  the  main  an  extension  and  simplification  of  the  one 
there  given. 

128.  General  Relation  between  Shear  and  Bending  Moment  in  Beams. — The  shear 
is  the  summation  of  all  the  components  of  the  external  forces  on  one  side  of  the  section  taken 
parallel  to  that  section.  The  bending  moment  is  the  sum  of  the  moments  of  all  the  external 
forces  on  one  side  of  the  section  about  the  centre  of  gravity  of  the  section.  There  follows  at 
once  from  these  this 

Proposition  :  The  beiiding  moment  at  any  section  of  a  beam  or  truss  is  equal  to  the  bend- 
ing moment  at  any  other  section  of  the  beam  or  truss  plus  the  shear  at  that  section  into  its  arm, 
plus  the  prodticts  of  all  the  intervening  external  forces  into  their  respective  arms. 

Thus  in  Fig.  186  we  have 

M^+,^  M^-^S^a-  Pz  (6) 

If  these  sections  be  taken  very  near  together,  and  the  interven- 
ing load  omitted,  so  that  in  eq.  (6)  a  =  dx  and  P  —  o,  then 
Mj^jf-a  becomes  M^Jrcixi  and  we  have 

M^.j^i^  =  M^-^  S^x    or   M^+j^  -  M^  =  Sdx. 


p 

" —  v 

r 

Fig.  186. 


But 


Mjc  —  dM\  therefore  we  have 

dM  =  Sdx  or 


dM 
dx 


=  S: 


(7) 


vice  versa,  w 


hen  the  shear  is  zero  the  bending  moment  is  constant. 


that  is  to  say,  the  shear  at  any  section  is  the  first  dif?erential  coefificient  of  the  bending  moment 
at  that  section.    If  the  bending  moment  is  constant,  therefore,  the  shear  must  be  zero,  and, 

But  when     —  =  o  the 
dx 

bending  moment  is  either  a  maximum  or  a  minimum  ;  therefore  when  the  bending  moment 
passes  through  a  maximum  or  a  minimum  the  shear  is  zero,  and  also  when  the  shear  becomes 
zero  the  bending  moment  is  either  a  maximum  or  a  minimum.  A  knowledge  of  this  relation 
can  often  be  used  with  advantage  in  the  analysis  of  trusses. 

129.  The  Deflection  of  Beams. — Let  Fig.  187  represent  a  portion  of  a  bent  beam  one 
unit  in  length.  The  sections  which  were  parallel  and  normal  before  bending  would  now  meet 
if  extended.    From  similar  triangles  we  have 

e  1 


But 

also 

and  we  have 


_      unit  stress      /  / 

E  =  — -.  —  =  —    or    €  =  --, 

unit  strain      e  /j 

I 

,^    //      '      M^y, . 

^o  =  -r,    or  f=—T-\ 


hence    e  = 


EI' 


r  ~  EV 


(8) 


Fig.  187. 


I  dxd^y 

But  from  the  calculus  we  have  -  =  — — -.    In  the  case  of  the  deflection  of  beams  the 

r  {(11) 


MODERN  FRAMED  STRUCTURES. 
MOMENTS,  STRESSES,  AND  DEFLECTION  OF  BEAMS. 


The  Beam  and  its  Load  with 
the  Moment  and  Shear 
Diagrams. 


M  mat 


FiG.  i88. 


 ^ 


2  (P 
Ma;M„ 


Fig.  igo. 


Moment  Equation  and 
Maximum  Moment. 
Mx  and  Mmax. 


Mr  =   —  PX 


AJmax.  =  -  PI 


2 


F 

2 


.  ,         _  PI 


2 


^ max,  — 


Pig.  193. 


Mx=^ 


X>  Zi 

PZ'iX 

Mx  =  — ^  P{x-z{) 


Mmax.  = 


PziZi 


Equation  of  Elastic  Line, 
and  Maximum  Deflection  in 
terms  of  the  Loading;. 


Pl^ 


A  = 


8^7 


J  = 


Px 
48^ 


384^5/ 


jr  <  zi 


Pz^(l—x) 


y-- 


bEll 
Pz 


for  jr  =  i|/3[z,{2z5-|-3,)] 


Maximum  Deflection  in 
terms  of  Stress  on  Ex- 
treme Fibre  of  Sym- 
metrical Sections. 


2/n 


2Eh 


fJl 


24  Eh 


+ 


I-*! 


PtJNDAMENTAL  RELATIONS  IN  THE  THEORY  OF  BEAMS. 
MOMENTS,  STRESSES,  AND  DEFLECTION  OF  BEAMS. 


The  Beam  and  iis  Load  with 
the  Moment  and  Shear 
Diagrams. 


®  ® 


Moment  Equation  and 
Maximum  Moment. 
Mx  and  Mmax. 


M  M 


Sibfij  I  s-0 


S2 


Fig.  193. 


S,l-Ri 


JS2 


Fig.  194. 


Fig.  195. 


X  <  z 
Mx  =  Px 
x>  z 
M=  Pz  —  Mmax. 


X  <  Zi 

Mx  --  Pi{l-x)-P{z2  -  x) 

X  >  Zi 

Mx  =         -  x) 
Mmax.  =  P\{1  —  Za) 

for  a:  = 


^1  =  I// 


4 


^moj:.  —  — 


for    =  o 


Equation  of  Elastic  Line, 
and  Maximum  Deflection  in 
terms  of  the  Loading. 
y  and  A 


X  <.  z 

y  =  6^  [3^^  -  3z*-  x^] 


x>  z 


^  = 


<  0j 


4-  ^Pz-iX-*  -  Z'.j^] 


^   >  22 


+  -iPz^^x  -  /'Zj'] 


48^/' 


.(/  -  Xt^l  -  2X) 


J  =  0.0054 


EI 

for  j:  =  0.578/ 


Maximum  Deflection  in 
terms  of  Stress  on  Ex- 
treme Fibre  of  Sym- 
metrical Sections. 
A, 


'-P-  z» 


It) 


0.0864//' 

Eh 


Maximum 

Stress  on  Ex- 
treme Fibre  in 
terms  of  the 
Loading.  Sym- 
metrical Sec- 
tions. 
/■ 


Pzh 
^  ~  2l 


/= 


ibl 


I  dy 

deviation  from  a  horizontal  line  is  so  small  that  we  may  call  dl  =  dx  \  hence  -  =  -j— „,  and  we 

r  dx 

have,  since  the  bending  moment,  M,  is  always  equa'  to  the  moment  of  resistance,  , 


r~E/''dx" 


(9) 


which  are  the  fundamental  equations  in  the  deflection  of  beams. 

d'y  M 

From  the  differential  equation        =       we  can,  by  giving  to  M  its  value  in  terms  of  x, 

dy  . 

and  integrating  once,  obtain  =  t,  the  angle  the  beam  makes  with  the  horizontal.  By 
integrating  again  we  obtain  y,  the  vertical  deflection  of  the  beam  at  any  point  from  its  normal 


MODERN  FRAMED  STRUCTURES. 

position.  The  modulus  of  elasticity  E  and  the  moment  of  inertia  /  are  usually  constant. 
The  latter  may  be  made  to  vary  as  some  function  of  x,  and  the  integration  made  with  / 
variable.  These  problems  are  worked  out  in  detail  in  books  on  applied  mechanics,  and  only 
a  summary  will  be  given  here  of  the  results  for  the  more  ordinary  cases.  In  solid  beams  and 
m  plate  girders  the  deflection  due  to  shear  is  neglected,  though  it  doubtless  is  something 
appreciable.  These  equations  do  not  apply  to  framed  structures,  since  no  account  is  taken  of 
the  deflection  due  to  strains  in  the  web  system,  which  in  ordinary  bridges  is  nearly  equal  to 
that  from  the  chords.    (See  Chap.  XV.) 

130.  The  Distribution  of  Shearing  Stress  in  a  Beam.— It  is  proved  in  mechanics 
that  wherever  a  shearing  stress  acts  along  any  plane  in  an  elastic  solid,  there  is  another  shear- 
mg  stress  of  the  same  intensity  at  that  point  acting  on  another  plane  at  right  angles  to  the 
first.  Also,  that  the  effect  of  these  two  equal  shearing  stresses  at  right  angles  to  each  other 
is  to  produce  two  direct  stresses,  of  the  same  intensity,  also  at  right  angles  to  each  other,  and 
at  angles  of  45°  with  the  former  planes,  one  of  these  direct  stresses  being  tension  and  the 
other  compression,  as  shown  in  Fig.  196. 

The  general  equation  showing  the  value  of  the  intensity  of  the  shearing  stress  at  any 
point  in  the  cross-section  of  a  beam  of  any  form  is 

,  S 


1  J]  ybdy*  (10) 

'>-5^&  CAcrcA's  /f/ecA  /)r7  Jd'S 
where  q'  ~  intensity  of  shearing  stress  on  any  plane;  r  o  df  ^  dp  /h  Soi{/-i,af^ 

S  =  total  shearing  stress  on  the  section  ;  ~~~ 
/=  moment  of  inertia  of  the  section  about  its  neutral  axis; 
b'  —  breadth  of  section  where  shearing  stress  —  q'  \ 

y  —  distance  of  the  plane  where  shearing  stress  is  q' ,  from  the  neutral  axis  of  the  sec- 
tion ; 

J',  =  distance  of  extreme  fibre  on  that  side  of  the  neutral  axis,  from  that  axis; 
h  —  breadth  of  section  at  distance  7  from  neutral  axis. 


Whence  it  follows  that  the  integral  fj'ybdy  is  the  statical  moment  of  the  area  outside 

the  longitudinal  plane  on  which  the  shearing  stress  is  taken,  about  the  neutral  axis  of  the 
beam.    We  may  therefore  define  the  intensity  of  the  shearing  stress  as  follows : 

The  intensity  of  the  shearing  stress  at  any  point  in  a  beam  of  solid  section  of  whatever  form 
ts  equal  to  the  total  shearing  force  on  the  entire  cross  section  multiplied  by  the  statical  moment  of 
the  area  of  the  section  outside  the  longitudinal  plane  of  shear  tn  question  about  its  axis  in  the 
7ieutral  plane,  divided  by  the  product  of  the  amount  of  inertia  of  the  entire  section  into  the 
breadth  of  the  section  at  that  point. 

From  this  there  may  be  deduced  the  following  relations  for  special  cases  :  For  solid 
rectangular  sections  the  intensity  of  the  shear  is  zero  at  the  top  and  bottom  sides  of  the  beam, 
and  increases  towards  the  centre  as  the  ordinates  to  a  parabola  having  its  axis  coincident  with 
the  neutral  axis  of  the  beam.  Hence  the  maximum  shearing  stress  is  found  at  the  centre  of 
the  section,  where  its  value  is  f  of  the  mean  intensity,  or  the  shear  at  the  centre  of  a  solid 
35 

rectangular  beam  is  —r,  as  shown  in  Fig.  196.    If  this  beam  is  subjected  to  a  uniform  load 

the  total  shear  is  zero  at  the  centre  and  increases  uniformly  to  the  ends.  The  total  shear  at 
any  section,  as  5, ,  S^,  S,,  etc.,  is  shown  on  the  lower  diagram  as  the  length  of  the  correspond- 
ing vertical  ordinate.  In  the  upper  figure  this  same  total  shear  appears  as  the  area  of  the 
parabola  drawn  at  that  section,  horizontal  ordinates  to  which  represent  the  shear,  at  the 


*  Rankine's  Applied  Mechanics,  §309,  eq.  (i). 


FUNDAMENTAL  RELATIONS  IN  THE  THEORY  OF  BEAMS.  13S 


corresponding  point,  on  a  surface  one  unit  long,  having  the  breadth  of  the  beam, 
being  either  vertical  or  horizontal,  and  normal  to  the  plane  of  the  paper.  Since 
ordinate  to  a  parabola  at  its  vertex  is 


this  surface 
the  middle 


I  the  mean  ordinate,  it  follows  that 

This  may  be  derived  from  equation 
(10)  directly,  by  taking  b  constant  for  a 
rectangular  section,  whence  we  have 


V  — ✓ 
»□» 

C  T 

^'  2A 

/  *2  2A 

Si  s 

Fig.  196. 


h 

for  /  =  o,  and  y,  —  -. 

For  a  plate  girder,  eq.  (10)  gives  directly  the  shearing  stress  at  any  point.  In 
shown  in  Fig.  197,  let  the  web  be  f  in.  thick  and  48  in.  high,  while  the  flanges  are 

Uniformly  Loaded 


the  girder 
I  in.  thick 


I2        <lv^  qArQ'O- 


~       On  bo 


Fig.  ig7. 

and  12  in.  wide.    Then  the  neutral  axis  lies  at  the  centre  of  the  girder,  the  area  of  one  flange 

is  9  sq.  in.,  and  its  statical  moment  about  the  neutral  axis  is  219.4  inch-pounds.    The  moment 

of  inertia  of  the  entire  cross-section  is  15,456.    Whence  we  have  for  the  intensity  of  the  shear- 

S                       I      219.4  \ 
ine  stress  at  the  inner  planes  of  the  flanges,  ^,  =  ^  /,  ybdy  =  S\  =0.0011 85 

^  ^  t>     •  V.       Ily^Jy'    ^  \I2  X  15.456/ 

(     219.4  N 

per  sq.  in.,  while  in  the  outer  sides  of  the  web  we  have     =  S  ~  0.0378^  per  sq.  in. 

S    p'  I     327.4  \ 

The  intensity  of  stress  at  the  neutral  axis  is  ^0  —  Yt  ^     iy^^^f)  —     \|~x"     ^^^j  ~  O.05655 

per  sq.  in. 

These  three  intensities  of  shearing  stress  are  indicated  in  the  "  intensity  "  shear  diagram 
in  Fig.  197,  plotted  to  the  left  of  the  plane  of  shear.    The  value  of  the  average  shear- 

ing  intensity  when  the  web  is  assumed  to  carry  it  all  is  ^  =  3  —  0.05656".    This  is  so 


I  X  48 


near  the  value  of  ,  the  true  maximum  shearing  stress,  that  the  ordinary  assumptions  of 
web  taking  all  the  shear,  and  thus  being  uniformly  distributed  over  the  web,  are  seen  to  be 
justified. 

By  taking  qb  as  the  .  total  shearing  stress  on  the  lamina  bdy,  we  may  construct  the  total 
shear  diagram  for  this  plane,  as  plotted  to  the  right  of  the  plane  of  shear  in  Fig.  197.  In  this 
diagram  the  total  shear  in  a  horizontal  plane  is  seen  to  be  the  same  on  the  interior  sides  of  the 
flanges  and  on  the  outer  edges  of  the  web,  and  the  area  of  the  diagram  represents  the  total 
shear,  5,  on  the  section. 


MODERN  FRAMED  STRUCTURES. 


It  is  important  to  note  that  since  the  shearing  stress  is  nearly  constant  across  the  section 
the  resulting  direct  stresses  in  the  web,  which  are  of  equal  intensity  with  the  shearing  stress, 
are  also  nearly  uniform  in  amount  across  this  section,  not  varying  from  zero  at  the  top  and 
bottom  fibres  to  f  the  mean  at  the  centre,  as  in  the  case  with  plane  rectangular  sections. 

If  the  beam  be  loaded  at  the  centre,  or  with  two  concentrated  loads,  as  in  the  case  of  a 
railway  bridge  floor-beam,  then  the  shear  is  constant  on  the  outer  ends  of  the  beam,  and  hence 
over  this  portion  the  web  is  subjected  to  nearly  equal  shearing  and  direct  stresses.  The  com- 
pressive stresses,  acting  at  45°  with  the  vertical,  and  in  a  direction  downwards  towards  the 

end  supports,  tend  to  buckle  the  web  plate.  Taking  a 
strip  aa',  Fig.  198,  the  intensity  of  the  compressive  load  at 
the  ends  of  this  strip  in  pounds  per  square  inch,  if  the 
dimensions  of  the  cross-section  be  the  same  as  taken  above, 
would  be  0.03785,  while  at  the  centre  it  would  be  0.05656". 
If  there  were  no  tensile  stress  at  right  angles  to  this  strip, 
tending  to  hold  it  into  its  true  plane,  the  strip  should  be 
dimensioned  in  thickness  to  carry  this  unit  load  0.0385  as 
a  load  on  a  column  of  that  length,  which  is  1.4  h. 

This  kind  of  analysis  would  give  an  extravagant  thickness  of  metal.  Just  what  the 
restraining  influence  of  the  tensile  stress  is  cannot  be  determined  theoretically,  and  no  ade- 
quate experiments  have  ever  been  made  to  show  it  empirically. 

Aside  from  the  strengthening  influence  of  the  tensile  stress  in  the  web  it  is  common  to 
further  stiffen  it  against  buckling  by  riveting  angle-irons  on  the  web  in  a  vertical  position,  as 
shown  on  the  left  end  of  the  beam  in  Fig.  199.    On  the  right  end  these  "  stiffeners"  are 


s' 


Fig.  ic 


00000 


00000 


O  O  O  Q  O 


00000 


0000  0000000000 


000000  OQOOOOOOOO 


Fig.  igg. 

placed  in  an  inclined  position,  directly  opposed  to  the  compressive,  or  buckling,  stresses.  In 
this  position  they  are  much  more  efficient  in  resisting  the  only  strains  in  the  web  which  are  at 
all  likely  to  cause  failure,  and  in  deep  girders  they  might  be  placed  in  this  position.* 

CONTINUOUS  GIRDERS. 

131.  The  Continuous  Girder  is  very  seldom  employed  in  this  country  except  in  swing 
bridges.  The  greatest  objection  to  its  use  is  the  uncertainty  in  the  stresses  resulting  either 
from  a  settlement  in  the  supports  or  the  impossibility  of  making  these  fit  exactly  to  the  nor- 
mal or  unstrained  profile  of  the  girder.  A  great  deal  of  literature  can  be  found  in  standard 
works  on  this  subject,  the  original  contributions  of  greatest  value  being  Clapeyron's  Theorem 
of  Three  Moments  (1857),  Weyrauch's  adaptation  to  concentrated  loads  and  unequal  spans 
(1873),  and  Merriman's  simplification  of  the  same  (1875). 

In  Fig.  200  {a)  and  {b)  two  cdnsecutive  spans  are  taken  from  a  series  of  an  indefinite 
number,  these  two  being  loaded  uniformly  in  the  one  case  with  the  unit  loads       and  p^,  and 


Mr-l 

I  


Pr-l 


Sr-l 


-l-fzrr 


nsT" 


Mr+l 
1 

1 


Mr-l 


A 


-tr- 


Mr 


r+1 
1 


Ir'-r 


■l-r  


(6) 


Fig.  200. 


*See  a  full  discussion  of  this  question  by  the  author  in  Engineering  News  of  April  25,  1895  (Vol.  XXXIII,  p.  276). 


FUNDAMENTAL  RELATIONS  IN  THE  THEORY  OF  BEAMS.  137 


in  the  other  with  the  two  concentrated  loads  and       ,  respectively.    Then  from  the 

Proposition  in  Art.  128,  and  eq.  (6),  we  may  write  at  once,  for  the  moment  at  any  section  in  the 
{r  —  i)th  span,  di.stant  x  from  the  left  support, 

=  M^.,  +  Sr-^X  -  y>r-^x'  (13) 

For  a  concentrated  load         distant  a  =  kl,.,  from  the  left  support  we  have 

=  M^_,  +  S^.^x  -  P^.ix  -  (I3«) 

For  more  than  one  load  in  the  span  the  sign  of  summation  is  inserted  before  the  term  in 

d'y 

Pr-i  •    From  these  equations,  and  the  general  differential  equation  of  the  elastic  curve, 

M  dy 

—        we  may  integrate  once  and  find  -7-  =  tan  t  at  each  end  of  the  (r  —  i)th  span,  by  mak- 
E/  ux 

ing  X  —  o  and  then  x  =  /,._,.    By  changing  the  subscripts  these  results  would  also  apply  to 

the  rth  span,  in  which  case  the  t^  (or  x  =  o  would  equal  the  t^.,  for  x  —       .    By  equating 

these  two  values  of  /  we  obtain  for  uniform  loads 

^...4..  +  2j/.(4..+  /.)  +  iIf,^,/.z=  -K/>.-.4-.'+A4').      ....  (14) 

while  for  concentrated  loads  in  these  spans  we  have 

+         +  /.)  +       =  -  2p^-A.,\^ -  2P^i;{2k  -  Ik'  +  k\  (14^) 

no  allowance  being  made  for  any  settlement  of  supports. 

These  are  two  general  forms  of  Clapeyron's  famous  Equation  of  Three  Moments,  first  pub- 
lished in  "Coniptes  Rendus,"  Dec.  1857.  The  forms  here  given  are  due  first  to  Weyrauch, 
Leipzig,  1873,  ^"d  to  Merriman,  Phil.  Mag.,  1875. 

When  the  spans  are  all  equal  and  load  uniform  over  the  whole  bridge,  the  equations 
become  very  much  simplified,  and  are  as  follows : 

=  m,^aM,  +  m, 


=  M„.,  +  4M„_,JrM„,  J 

When  the  entire  series  of  equations  of  three  successive  moments  are  written  for  any 
series  of  spans,  the  number  of  these  is  always  two  less  than  the  number  of  supports,  or  than 
the  whole  number  of  M'?,  found  in  the  equations.  But  the  end  M's  are  always  zero,  which 
makes  the  number  of  unknowns  equal  to  the  number  of  equations,  and  hence  all  the  inter- 
mediate M's  can  be  found.  The  algebraic  reduction  is  somewhat  tedious  and  will  not  be 
given  here.  After  the  M's  are  all  obtained,  the  shears  can  be  found  by  means  of  eq.  (13),  or 
(13a),  and  hence  the  supporting  forces. 

Having  found  all  the  external  forces  acting  upon  the  beam,  the  moment  and  shear  at  any 
section  can  be  found  by  the  ordinary  methods  as  readily  as  for  a  simple  beam  on  two  sup- 
ports. The  great  value  of  "  the  equation  of  the  three  moments  "  consists  in  its  enabling  u? 
to  find  the  supporting  forces. 

The  following  formulce  apply  only  to  beams  and  trusses  having  a  uniform  moment  of  inertia 
spans  all  of  equal  length,  and  when  no  settlement  occurs  at  the  supports.* 

I.  Bending  Moments  at  the  Supports. 

(A)  For  Uniform  Load  over  the  rth  Span,  Spans  all  Equal.  —  The  moment  at  the 
M th  support,  counting  from  the  left,  for  any  number  of  spans  wholly  loaded  with  the  uniform 


(15) 


*  This  condition  is  usually  stated  as  "supports  on  a  level."  This  is  very  misleading,  as  the  formulae  do  apply  to 
supports  out  of  level,  provided  they  are  fitted  to  the  profile  of  the  beam  or  truss  in  its  normal  unstrained  condition. 


MODERN  FRAMED  STRUCTURES. 


load  p  per  foot,  the  subscripts  r  applying  to  all  loaded  spans,  n  being  the  whole  number  of 
spans, 

//'    r  1 

(For  loaded  spans  on  left.)       ( For  loaded  spans  on  right.) 

(B)  For  Concentrated  Loads  in  the  xth  S-^an,  Spans  all  Equal.  —  The  moment  at 
the  wth  support,  for  loads  P,  at  distances  a  =  kl  from  left  of  the  rih  span  or  spans,  when  total 
number  of  spans  =  n,  all  equal,  is 

M„  =  -  7— i— -f 2\2P{2k  -  ik-^        +  :^P{k  - 

''n  -  I     \     4^  n  L 

l,For  loaded  spans  on  left  of  wth  support.) 

+  :2\:^p{2k  -  ik^  +  ^P{k  -  (17) 

(For  loaded  spans  on  right  of  ;«th  support.) 

If  both  uniform  and  concentrated  loads  are  found  upon  the  same  span,  then  both  formulae 
must  be  used.  When  more  than  one  span  is  loaded,  the  data  mu.st  be  worked  out  for  each, 
and  the  sum  taken,  as  indicated  by  the  primary  signs  of  summation.  The  secondary  summa- 
tion signs  for  concentrated  loads  signify  the  summation  for  the  several  joints  or  concentrated 
loads. 

The  following  values  are  to  be  used  for  the  c  coefficients: 

f ,  o  =  —      56  =  —  10,864 

^,  =  +   I  =  +    209  =  +  40,545 

^3=-   4  =  -    780  =  -  151,316 

^  =  +  15  =  +  2,911  =  +  564,719 

following  the  law  that  c^  =  —  4^m-i  ~  ^m-a- 

^     Mm-.l  Mm  Mm+1  ^ 

'ffs,       i,J    fts...    ffs.      t      I  t         f  f 


— Ir—^  h-  1^-^~It  V—'tn, — ^ 

Fig.  20I. 

Since  the  concentrated  load  is  usually  at  a  panel  point,  if  there  were  five  panels  in  the 
loaded  span,  k  would  be-|^,f,  |,  and  |,  for  loads  at  the  first,  second,  third,  and  fourth  panel  points, 
respectively.  The  annexed  table  of  the  values  of  the  two  terms  in  k,  viz.,  {k  —-  k^)  and 
{2k  —  Tfk^  -\-  k'),  has  been  compiled  for  a  more  ready  computation  of  these  quantities.  They 
are  computed  for  the  aliquot  parts  of  a  span-length,  or  for  the  joints  of  a  truss  of  equal  panel 
lengths,  for  all  numbers  of  panels  up  to  twelve. 

When  we  have  but  two  equal  spans,  as  is  commonly  assumed  to  be  the  case  in  swing 
bridges,  equations  (16)  and  (17)  reduce  to  the  following: 

(C)  Ttvo  Equal  Spans,  Uniform  Load. — —  —        for  full  load  on  either  span ;  or 

when  both  spans  are  fully  loaded, 


^,=  -y  (18) 

(D)  Two  Equal  Spans,  Concentrated  Loads. 

M,=  -^^-2P{k-  k^)^'2P(2k-ik'-^k')).   ......  (19) 

(For  left  span.)  (For  right  span.) 


FUNDAMENTAL  RELATIONS  IN  THE  THEORY  OF  BEAMS. 
VALUES  OF  {k  —  k')  AND  OF  (2k  —  ^k'  +  k*)  AT  PANEL  POINTS.* 


139 


Two  Panels. 

Six  Panels. 

Nine  Panels. 

Eleven  Panels. 

/I  -  ,§3 

.;.=^ 

/J  -  i8 

k  —  k'^ 

■375 

1 

« 

8 

4 

¥ 
5 

.162 
.296 

•  375 

•  370 

•  255 

s 

4 

s 
J 

f 
1 

1 

1 

8 

f 
e 
7 

.110 
.211 

.296 

•  35f> 

•  384 

•  370 
■  307 
.186 

8 

I 
6 
? 
6 
1f 
4 
¥ 
3 
5 

% 
1 

t't 

2 

TT 

tV 

4 

TT 

5 
TT 

e 

TT 
7 
T  ( 

8 

TT 

.090 
.176 
•253 
.316 
.361 
•383 
.379 

•343 
.271 
.156 

f 

t't 

s 

/ 

Three  Panels. 

2k  -  3/J*  +  k'^ 

2k  —  3/^'  + 

t't 

1  0 

TT 

Seven  Panels. 

2k  -  jk^  +  k^ 

f 

.296 
■  370 

k  -  B 

Ten  Panels. 

Twelve  Panels. 

2k  -  3/^2  + 

f 

f 
5 

T 
8 

.140 
.263 
•350 
.851 

•349 
.228 

6 
T 
6 
T 

2 

1 
7 

k  - 

k  -  k^ 

Four  Panels. 

1 

3 

Iff 

6 

TT 
6 

TIT 
7 

T5 

8 

TO 

9 

TIT 

.099 

.192 

.273 
.336 

•375 
•384 
•357 
.288 
.171 

9 

TS 
8 
TT 

7 

Tff 
« 

TO 

T^IT 

T% 
3 

TS 

T% 
1 

1 

TI 

\\ 

T% 

5 
TT 

6 

7« 

T? 

t\ 
tV 

1  0 

T3 
1  1 
T2 

.082 
.162 
.234 
.296 

•  344 
•375 

•  385 

•  370 
.328 

•255 
.  146 

1  1 
TJ 
I" 
T5 
9 

TS 

tV 

tV 

« 
TJ 

G 
TJ 

t\ 

3 

tV 

,v 

k  -- 
-/ 

k  - 

zk  -  + 

i 
f 
i 

.234 

•  375 
.328 

f 

T 
i 

Eight  Panels. 

2k  -  3/^2  +  k"- 

2k  -  3i»  +  ySrS 

k  -  1^ 

7 

« 

a 

2 
1 

Five  Panels. 

J 

2 
If 
8 

1 
5 

1 
7 

s 

.123 
■  234 
.322 
•375 
.381 
•  328 
.205 

2k  -         +  P 

/ 

^  -  /J« 

1 
3 
3 
4 

.  192 
•33'^ 
.384 
.288 

4 
E 

1 

a 
1 

2k  —  3/^-^  •+  4^ 

/ 

2k  -  3^'  + 

(E)  i^i^r  Uniform  Load  over  the  Entire  Girder,  Spans  all  Equal. — Equation  (16)  now 
reduces  to 


[' 


J12 ' 


(20) 


This  formula  is  evaluated  in  the  following  diagram  for  all  supports  for  girders  ot  ni.i^. 
spans  or  less. 

*  Condensed  from  Prof.  Malverd  A.  Howe's  "Theory  of  the  Continuous  Girder,"  where  the  student  will  find  the 
most  extended  theoretical  treatment  of  the  continuous  girder  in  the  English  languaf»e. 

f  This  and  the  following  equation,  (21),  are  taken  from  Du  Bois'  "  Framed  Structures,"  with  some  change  of  form. 


14° 


MODERN  FRAMED  STRUCTURES. 


MOMENTS  AT  SUPPORTS;  TOTAL  UNIFORM  LOAD;  SPANS  ALL  EQUAL. 
COEFFICII'.NTS  OF  (— 


0 

I 

8 

2 

0 

A 

A 

A 

0 

I 

I 

10 

2 

1 

10 

J 

0 

A 

A 

A 

A 

0 

I 

3 

28 

2 

2 

28 

3 

28 

A 

0 

- 

A 

A 

A 

A 

A 

0 

I 

4 

38 

2 

3 

38 

•J 

3 

38 

4 
38 

e 

A 

A 

A 

A 

A 

A 

0 

I 

II 
104 

2 

8 

104 

•X 

J 

9 

104 

8 

104 

e 

0 

II 
104 

6 

0 

A 

A 

A 

A 

A 

A 

A 

o 

I 

2 

II 

12 

A 

12 

c 

J 

II 

6 

15 

7 

142 

142 

142 

*+ 

142 

142 

142 

/ 

A 

A 

A 

A 

A 

A 

A 

A 

I 

41 

2 

30 

33 

32 

33 

6 

30 

41 

8 

388 

388 

3 

388 

4 

388 

5 

388 

388 

7 

388 

A 

A 

A 

A 

A 

A 

A 

2 

41 

45_ 

44 

44 

6 

45 

41 

8 

530 

530 

3 

530 

4 

530 

5 

530 

530 

7 

530 

530 

A 

A 

A 

A 

A 

A 

A 

A 

In  general  it  may  be  said  that  the  moments  at  the  supports  next  to  the  ends  are  always 
the  greatest,  and  are  there  about  xV/*^'  >  ^^^^  they  are  least  at  the  third  supports  from  the 
ends,  where  they  are  about  -j^/" ;  and  near  the  centre  they  are  nearly  uniform  at  about  tV/*^  • 


II.  Shears  at  the  Left  Ends  of  the  Spans. 

Having  found  the  moments  at  all  the  supports  for  the  particular  loading  in  question,  it 
remains  to  find  the  shears  on  the  left  ends  of  each  span,  when  the  stresses  in  all  the  members 
are  readily  computed.  To  find  these  shears,  make  x  =  in  eqs.  (13)  and  (i3<2),  when  they 
become 

for  a  uniform  load  over  the  (r  —  i)th  span,  and 

=  M^.,  +  -  2P,.J^_li  -  k) 

for  a  series  of  concentrated  loads  in  the  {r  —  i)th  span  ;  whence,  for  the  shear  at  the  left  end 
of  the  rth  span,  or  at  the  right  of  the  rth  support, 

M    —  M 

Sr=  '  +  hplr   (21) 

as  the  shear  at  the  left  end  of  any  span  which  is  uniformly  loaded,  and 

M     —  M 

s^  =  -^^-^:2P{x-k)    (2i«) 

'■r 

as  the  shear  at  the  left  end  of  any  span  carrying  concentrated  loads. 


FUNDAMENTAL  RELATIONS  IN  THE  THEORY  OF  BEAMS. 


For  unloaded  spans  the  second  terms  in  the  right  members  become  zero. 
Evidently  the  shear  at  the  right  end  of  any  span  is  that  on  the  left  end  minus  the  inter- 
vening load. 

The  following  gives  the  shears  on  each  side  of  the  supports  for  the  case  of  uniform  load 
over  the  entire  girder.    The  supporting  force  is  the  sum  of  the  two  shears  at  that  support. 

SHEARS  AT  SUPPORTS  ;    TOTAL  UNIFORM  LOAD  ;   SPANS  ALL  EQUAL. 
COEFFICIENTS  OF  (W). 
I 

o  1  I  I  I  o 

2  2 
I  2 

o  I  3  S  I  5  3  I  o 

8  8  8 

 I  2  3 

o  I  4  6jJ  £76  4l  o 

lO  TO  lO  10 

1  2  3  4 

o  I  II         17  I  IS         13  I  13         15  I  17         II  I  o 
28  28  28  28  28 

 I  2  3  4  5 

o  I  15         23  I  20         18  I  19         19  I  18         20  I  23         15  I  o 
38  38  38  38  38  ~  38 

I  2  3  4  5  6 

o  I  41         63  I  55         iil^         53  I  53  51  I  49  55  I  63         41  I  o 

104  104  104  104  104  104  104 

I  2  3  4  5  6  7  

o  I  56         86  I  75         67  I  70         72  I  71  71  I  72  70  I  67  75  I  86  56  I  o 

142  142  142  142  142  142  142  142 

 I   2  3^  J  5^  6  7  8  

o  I  IS3      235  I  205      183  I  191       197  I  195      193  I  193      195  I  197      191  I  183      205  I  235      153  I  o 
388  388  388  388  388  388  388  388  388 

I  2  3  4  5  6  7  8  9 

0  I  209      321  I  280      250  I  261      269  I  266      264  I  265      265  I  264      266  I  269      261  I  250      280  I  321      209  I  o 
530  530  530  530  530  530  530  530  530  530 

III.  COMPUTATION   OF  STRESSES  IN  THE  MEMBERS. 

From  the  moments  at  the  supports  and  the  shear  on  the  right  of  each  support  all  the 
stresses  are  readily  found.  Thus  for  any  section  distant  x  from  the  left  end  of  the  wth  span, 
we  have,  from  eqs.  (13)  and  (13^;), 

M^„,,)  =  M„,^  S„,x -\p,,x'  (22 j 

for  a  uniform  load  in  this  span,  and 

^(«,.)  =       +  S„,x  -  kPJ^x  -  kQ  {22a) 

o 

for  concentrated  loads  in  the  ;«th  panel 

If  there  are  110  loads  in  this  span,  then  the  last  term  disappears  from  each  equation. 
Also,  for  shear  at  any  section  distant  x  from  the  left  end  of  the  wth  span,  we  have 

S(,„^)  —  S,„  —  p,„x  (23) 

for  uniform  load,  and 

S(,„^-)  =  S,„  —  2P  (2^a) 


142 


MODERN  FRAMED  STRUCTURES. 


for  intervening  concentrated  loads.  Or  in  other  words,  the  shear  at  any  section  is  equal  to 
the  shear  at  any  other  section  minus  the  intervening  external  forces. 

Having  the  moments  and  shears  at  any  section,  the  stresses  in  the  members  are  readily 
found. 

Note. — For  a  complete  discussion  of  the  theory  of  the  continuous  girder  with  varying  spans  and 
moments  of  inertia,  see  "  The  Theory  of  the  Continuous  Girder"  (120  pp.),  by  Prof.  Malverd  A.  Howe,  Engineer- 
ing News  Publishing  Co.,  1889.  For  a  complete  discussion  for  constant  moments  of  inertia,  see  Prof.  Merri- 
man's  original  paper,  Van  Nostrand's  Science  Series,  No.  25  (1875),  and  also  Prof.  Du  Bois'  "  Fratned 
Structures."  For  a  complete  graphical  analysis  of  the  problem,  moment  of  inertia  constant,  see  Prof.  Eddy's 
"Researches  in  Graphical  Statics,"  V&n  Nostrand,  1878;  also  the  same  inserted  in  Prof.  Church's  "Mechanics 
of  Materials,"  and  in  Prof.  Burr's  "Bridges."  The  continuous  grider  is  now  so  little  employed  in  America 
that,  in  the  opinion  of  the  authors,  it  is  no  longer  necessary  to  teach  the  details  of  this  practice  in  our  en- 
gineering schools. 


COLUMN  FORMULA. 


143 


CHAPTER  IX. 

COLUMN  FORMULAE. 

132.  Crushing  Strength. — Engineering  materials,  when  tested  in  compression,  divide 
themselves  into  two  very  distinct  categories  :  those  which  fail  absolutely  by  crushing  to 
pieces  along  diagonal  planes,  and  those  which  merely  distort,  or  flow,  under  the  increasing 
load,  and  never  disintegrate  under  any  pressure,  however  great.  Cast-iron,  hard  cast-steel, 
stone,  brick,  cement,  and  the  like  are  of  the  former  class,  while  wrought-iron,  all  grades  of 
rolled  steel,  the  alloys,  and  timber,  are  of  the  latter.  With  this  class  of  materials  "  crushing 
strength  "  should  be  understand  to  mean  the  resistance  the  substance  offers  to  cold  flowing, 
or  to  permanent  distortion  under  a  crushing  load.  But  this  point  is  called  the  "  elastic  limit 
in  compression  ;"  therefore  for  semi-plastic  materials,  like  the  rolled  metals,  the  elastic  li}nit  is, 
or  should  be  regarded  as,  the  ultimate  strength*  This  has  sometimes  been  called  the  "  crippling 
strength,"  but  for  all  practical  purposes  it  should  be  regarded,  in  the  designing  of  structures, 
as  the  "  ultimate  strength." 

In  all  grades  of  rolled  iron  and  steel  the  elastic  limit  in  compression  is  practically 
identical  with  this  limit  in  tension,  and  hence  the  ''elastic  limit"  as  found  by  a  tension  test  of 
the  materials  may  be  regarded  as  the  "  riltimate  strength"  of  that  material  in  compression.  For 
very  short  columns  of  such  a  material,  which  are  perfectly  straight,  with  exact  centering  in 
the  testing  machine,  and  with  end  bearings  which  resist  lateral  movement,  as  square,  hinged, 
or  pin  ends,  it  may  be  possible  to  place  a  greater  load  upon  the  column,  but  its  length  is  then 
permanently  shortened,  and  the  chances  are  greatly  in  favor  of  its  giving  way  by  lateral 
deflection  for  want  of  perfect  fulfilment  of  one  or  more  of  the  conditions  named.  All 
recorded  tests  of  wrought-iron  and  steel  columns,  therefore,  which  show  an  "ultimate 
strength  "  greater  than  the  elastic  limit  of  the  material  should  be  considered  as  abnormal  and 
misleading,  and  should  be  given  no  weight  in  any  experimental  tests  of  the  correctness  of  any 
proposed  formula,  or  in  the  derivation  of  the  constants  of  a  formula  to  be  used  for  computing 
the  strength  of  columns. 

Unfortunately  the  elastic  limit  in  column  tests  has  seldom  been  observed,  and  notwith- 
standing the  great  number  of  tests  made  of  full-sized  members,  as  well  as  on  laboratory 
samples,  we  are  still  almost  entirely  devoid  of  data  from  which  to  derive  the  constants 
entering  into  any  rational  formula  for  the  strength  of  columns. 

THREE  METHODS  OF  COLUMN  FAILURE. 

133.  I.  By  Direct  Crushing. — If  the  column  is  short,  without  internal  stress,  perfectly 
straight,  of  uniform  size  and  strength,  all  its  filaments  having  the  same  modulus  of  elasticity 
and  the  same  elastic  limit,  the  centre  of  gravity  of  the  imposed  load  coinciding  exactly  with 
the  centre  of  gravity  of  the  cross-section,  then  all  the  longitudinal  elements  of  the  column  will 
be  equally  compressed,  all  will  come  to  their  elastic  limit  at  the  same  time,  and  all  will  distort 
alike.  This  distortion  will  continue  indefinitely  without  lateral  deflection,  the  material  simply 
spreading,  or  flowing,  under  the  imposed  load. 

This  is  evidently  a  purely  ideal  condition,  and  can  never  be  realized  perfectly  even  in  a 

*  See  paper  on  Compressive  Strength  of  Steel  and  Iron,  by  Chas.  A.  Marshall,  M.  Am.  Soc.  C.  E.,  Ti  ans.  Am.  So», 
C.  £.,  VoJ.  XVII,  p.  53.    Consijlt  Plates  X  and  XI  for  proof  of  the  above  statement. 


144 


MODERN  FRAMED  STRUCTURES. 


carefully  arranged  experimental  test,  to  say  nothing  of  the  conditions  obtaining  in  actual 
practice.  Here  the  "  ultimate  strength  "  should  be  regarded  as  the  "  elastic  limit,"  notwith- 
standing greater  loads  will  be  resisted  after  permanent  distortion  begins.  Evidently,  so  long 
as  the  column  does  not  deflect  sidewise,  its  strength  per  square  inch  is  independent  of  its 
length,  or  of  its  ratio  of  length  to  radius  of  gyration,  and  dependent  only  on  the  elastic  limit 

of  the  material.  For  various  lengths,  or  for  increasing  values  of  ^,  therefore,  failure  by  crush- 
ing only  would  show  a  constant  unit  strength  equal  to  the  elastic  limit. 

134.  II.  By  Crushing  and  Bending  Combined. — If  any  of  the  above-named  conditions 
are  not  fulfilled,  then  the  column  will  bend  somewhat  under  all  loads,  the  bending 
increasing  with  the  load,  and  the  concave  side  of  the  beam  at  the  elastic  limit  will 
be  subjected  in  general  to  compressive  stresses  from  three  causes : 

P 

First,  to  a  stress p  =       uniformly  distributed  over  the  section. 


Second,  to  a  stress  p 


,  _  Pvy,  _  Pvy^  _  pvy^ 


I 


,  where  v  is  the  eccentric  displace- 


FlG.  202. 

Third,  to  a  stress  p 


ment  of  the  load,  is  the  distance  of  the  extreme  fibre  from  the  centre  of  gravity 
of  the  cross-section,  and  r  is  the  radius  of  gyration  of  the  section.  Fv  would  be 
the  bending  moment  caused  by  the  eccentric  position  of  the  load. 

^^Q^i^  P  ^t  the  elastic  limit,  due  to  the  bending  of  the  column 

under  the  load  F,  f  being  the  elastic  limit  stress  for  that  material,  and  E  the  modulus  of 
elasticity.  This  is  found  as  follows  :  \i  A  =z  lateral  deflection  under  the  load  P,  then  the  bend- 
ing moment  from  this  source  is  PA.  Since  the  moment  at  any  point  is  equal  to  Fy,  where  is 
the  deflection  from  a  right  line  at  that  point,  the  bending  moment  diagram  may  be  considered 
a  parabola  for  such  bending  as  occurs  within  the  elastic  limit.  But  for  this  law  of  moments, 
which  is  the  same  as  obtains  in  a  uniformly  loaded  beam,  the  deflection  at  the  centre,  in  terms 

of  the  stress  on  the  extreme  fibre  from  bending  only,  is  at  the  elastic  limit  A  =  r-^^  , 

^      ■'  48  Ey^ 

where  f  —  p  =  P'     P"  —  total  bending  moment  stress  on  the  outer  fibre.    (See  table  p.  132, 

beam  uniformly  loaded,  h  being  taken  equal  to  2j/,.)    If,  for  convenience,  we  assume  -^-^  — 

no  appreciable  error  will  be  made.  But 


M=  PA  = 


10  Ey^ 


p"I- 


Also,  in  terms  of  the  stress  produced  on  the  outer  fibres,  M  =  — ;  therefore  we  have,  since 
r=Ar\ 


£1 


P{f-p)i' 

10  Ey^ 


or 


A  '  loE 


We  may  now  write,  as  the  total  stress  on  the  extreme  fibre  at  the  elastic  limits 


/  =  P+P'+P"  =P 


whence 


loE  \r) 

f 


COLUMN  FORMULA. 


MS 


where  f  —  elastic  limit  of  the  material  in  compression  ; 
V  =  eccentric  displacement  of  the  load  in  inches; 

=  distance  of  outer  fibre  from  centre  of  gravity  of  the  section  in  inches  r 
£  =  modulus  of  elasticity  ; 
/  =  length  of  column  ; 


r  =  radius  of  gyration  of  the  cross-section  in  inches,  = 


This  is  a  strictly  rational  formula,  with  no  purely  empirical  constants,  for  a  column  fret 
to  revolve  at  the  ends,  and  will  give  good  results.  It  must  be  solved  by  trial  since  /  is  found 
on  both  sides  of  the  equation.  If_;/,  =  |r,  which  is  about  the  ordinary  ratio,  and  if  34,000 
for  wrought  iron  and  42,000  for  mild  steel,  with  £  =  27,000,000  for  iron  and  28,500,000  for 
steel,  this  formula  would  become. 


For  Wrought-iron,       p  =  — — '  ,  .   (2) 


I  I  ii'  I  34.000 -  /  IIV 

~^  xr     2  70.000.000  \rl 


ir     270,000,000  \ri 

and 


42,000 

For  Steel,  p=  ^  .  ^-^;^5^ypy  (3) 

~^  y  285,000,000V/ 

These  would  apply  only  to  columns  pivoted  on  knife-edges.  They  would  not  apply  to 
"  round-ended  "  columns,  because  the  point  of  application  of  the  load  shifts  as  the  column 
bends.  They  would  not  apply  to  a  hinged,  or  pin-connected,  column,  because  the  resistance 
to  motion  here  is  very  considerable,  which  greatly  increases  the  strength  of  the  column  by 
preventing  lateral  deflection. 

vy 

The  second  term  in  the  denominator,       ,  must  include  all  effects  of  initial  bends,  or 

kinks,  in  the  column,  and  differences  in  the  moduli  of  elasticity  of  the  elementary  forms  of 
which  it  is  composed,  as  well  as  eccentric  position  of  load.  These  are  usually  unknown  func- 
tions, and  hence  this  term  cannot  commonly  be  evaluated.    If  this  term  be  omitted,*  and  an 

/  — / 

empirical  constant  coefficient  used  for  in  the  next  term,  we  have 

■+"(-.) 


which  is  commonly  known  as  Gordon  s,  or  Rankincs,  Formula.^ 

135.  HI.  By  Bending  alone. — If  all  the  conditions  named  in  I  are  fulfilled,  and  we 
assume  the  column  is  loaded  somewhat  inside  its  elastic  limit,  while  the  length  increases,  the 
column  will  remain  in  stable  equilibrium,  and  undeflected,  until  the  length  reaches  a  particular 
point,  when  it  will  bend.  When  the  length  is  less  than  this  critical  amount,  if  the  column 
were  bent  by  a  transverse  force,  it  would  straighten  itself  under  its  load,  when  this  deflecting 
force  is  removed.  But  when  the  length  has  reached  a  certain  limit,  it  will  no  longer  be  able  to 
straighten  itself  under  its  load,  hut  ivill  retain  any  particular  deflection  7vhich  may  be  given  to  it. 
It  is  then  in  unstable  equilibrium,  and  any  further  increase  of  load  will  cause  the  bending  to 

*The  authors  do  not  admit  the  legitimacy  of  these  changes,  but  they  make  this  supposition  here  only  to  show 
what  changes  would  be  necessary  to  obtain  Rankine's  formula. 

I  For  shoft  columns,  with  eccentric  loads,  see  note,  p.  153,  and  also  eq.  (11),  p.  455, 


146 


MODERN  FRAMED  STRUCTURES. 


increase  till  the  elastic  limit  is  reached,  when  failure  is  inevitable.  For  columns  pivoted,  or 
free  to  turn  at  the  ends,  this  limiting  length  for  a  perfectly  ideal  column  is 


(5) 


The  shortest  length  of  column  which  could  act  in  this  way  would  be  found  by  making 
p  as  large  as  possible;  or  when 

p  —f=  elastic  limit. 


we  have  as  the  minimum  length  which  can  fail  by  bending  only 


>  77- 


E 
7 


(6) 


This  lower  limiting  ratio  of  /  to  r  for  a  perfectly  ideal  column  is  about  100  for  wrought- 
iron  and  about  85  for  mild  steel.  A  spring-steel  column  might  fail  in  this  way  for  a  ratio  of 
/  , 

-  as  low  as  50. 
r 

For  the  perfectly  ideal  column,  therefore,  centrally  loaded,  failure  would  occur  by  methods 
I  and  III  and  never  by  method  II.  That  is,  the  strength  would  be  constant  and  equal  to 
the  elastic  limit  for  increasing  lengths  until  the  limiting  length  is  reached,  when  it  would  fail 

by  bending.    The  strength  of  the  ideal  column,  free  to  turn  at  the  ends,  where  -    is  greater 

than  this  limit,  or  when  failure  occurs  by  bending  alone,  is 

7T  EI     Tt'EAr'     71' £  A 


or 


n'E 

0" 


(7) 


This  is  called  Etiler  s  Formula*  and  is  derived  as  follows  : 

Let  OQ,  Fig.  203,  represent  a  column,  free  to  turn  about  its  end  supports,  of  such  a 
length  that  it  may  be  in  unstable  equilibrium  under  the  load  P.  That  is,  under  this 
load  the  column  just  begins  to  deflect,  and  will  under  a  constant  load  retain  any 
deflection  which  may  be  given  to  it,  within  the  elastic  limit  of  the  material.  The 
bending  moment  at  any  section  distant  x  from  the  origin  at  O  is  then 


M=  -  Py^EI  -^^ 


(8) 


from  the  fundamental  equation  for  the  deflection  of  beams.  Whence 


(Fy_ 


P 


I  j  Multiplying  each  side  of  this  equation  by  dy  when  x  is  the  independent  variable  anc 
.-.i.  integrating  once,  we  have 


Fig.  203. 


^dy 
\dx 


Contributed  to  the  Berlin  Academy  by  Euler  in  1759. 


COLUMN  FORMULA. 


147 


When 


dy 
dx 


0,7  =  J,  = 


ldy_ 
\dx. 


EI 
whence 


deflection  at  the  centre  ;  therefore  C  — 
P  /Wf  dy 

-m^^  -/)-  ox  dx^^ ^-  ^ 


A\  and 


y 


.  arc  sni     -\-  C. 


When  ;f  =  o,  7  =  o.  .•.  C  —  o,  or 


Therefore 


P 


y 


=  arc  sin  — 
EI  A 


(9) 


y  =  A       X   ( 


10) 


This  is  the  equation  of  the  elastic  line,  the  curve  being  a  sinusoid.    But  for  x  —     y  =  A, 


and  we  have  from  (9) 


2\J  EI~  2' 


or  P- 


n'EI 


or 


P 


n'EAr\ 

Ar  ' 


or  P  =  r^. 


Tt'E 

1 


00 


which  is  Euler  s  Formula. 

136.  The  Effect  of  End  Conditions. — When  the  ends  are  free  to  turn,  the  column 

bends  in  single  curvature,  as  in  Fig.  204.  When  both  ends  are 
fixed  in  position  it  takes  the  form  of  a  double  reversed  curve,  as 
in  Fig.  205.  Here  the  portion  lying  between  the  two  points  of 
inflection  acts  as  a  whole  column  on  knife-edges,  but  the  length 
/ 

of  this  portion  is  only  — . 


When  one  ead  is  fixed  and  the  other 


free  to  turn,  the  portion  analogous  to  Fig.  204  is  |/,  as  shown  in 
Fig.  206.    Therefore,  when  a  formula  has  been  derived  for  the 

/ 

first  case  it  can  be  used  for  the  other  two  by  putting  for  /,  -  for 

fixed  ends,  and  \l  for  one  end  fixed  and  the  other  free  to  move. 

Therefore  we  may  write  the  following  theoretical  formula 
for  these  several  conditions  : 


Fig.  204.     Fig.  205.     Fig.  206 


Name  of  Formula. 


For  Pivoted  Ends. 


For  Fixed  Ends. 


For  one  end  Pivoted  and 
one  end  Fixed. 


148 


MODERN  FRAMED  STRUCTURES. 


Unfortunately  neither  of  these  end  conditions  is  ever  found  in  practice.  The  nearest 
approach  to  a  pivoted  end  is  the  ordinary  pin  connection,  but  the  pin  so  nearly  fills  the  hole 
that  when  the  column  is  loaded  the  frictional  resistance  to  slipping  is  a  very  material  source 
of  strength  to  the  column  by  preventing  lateral  deflection.  The  strengthening  effect  of  the 
pin  is  greater  the  larger  the  ratio  of  its  diameter  to  the  radius  of  gyration  of  the  column,  but 
even  a  very  small  pin  well  oiled  gives  a  much  higher  test  for  long  columns  than  a  perfectly 
frictionless  or  knife-edge  bearing. 

The  nearest  approach  to  a  fixed  end  commonly  found  in  structures  is  a  squarely  abutting 
end  upon  a  rigid  or  fixed  base.  This  is  equivalent  to  a  fixed  end  for  short  lengths,  but  for 
long  lengths,  where  the  fibres  on  the  convex  side  come  into  tension,  the  square-ended  column 
no  longer  acts  as  a  fixed  end,  since  the  joint  cannot  usually  resist  tension. 

137.  A  New  Formula. — For  theoretically  perfect  columns  and  central  loading,  failure 
would  occur  by  methods  I  and  III,  and  along  the  lines  ABD,  Fig.  207,  for  pivoted  ends,  and 
along  AFH  for  fixed  ends,  if  these  conditions  could  be  perfectly  satisfied.*  Any  failure 
in  these  necessary  limitations,  either  as  to  the  column  itself,  its  loading,  or  its  end  bearings, 
would  result  in  lowering  the  maximum  unit  stress,  except  for  very  short  or  for  very  long 
lengths.  Unless  the  loading  is  very  eccentric,  failure  will  always  occur  on  very  short  lengths 
for  p  —  f  —  elastic  limit  of  the  material.  For  very  long  lengths  the  column  fails  almost 
wholly  by  bending,  so  that  here  the  only  significant  condition  is  the  value  of  the  modulus  of 

/ 

elasticity,  the  end  conditions,  and  the  ratio  — .    But  these  extremes  include  all  practical 

lengths  of  columns,  and  hence  we  may  say  that  the  actual  strength  of  a  given  column  may  be 
found  anywhere  within  a  field  of  considerable  width,  depending  on  a  number  of  indeterminate 
conditions.  Any  convenient  formula,  therefore,  having  its  locus  centrally  located  in  this  field 
of  experimentally  determined  results,  and  satisfying  the  theoretical  requirements  for  very  long 
and  for  very  short  columns,  where  the  unknown  functions  are  relatively  unimportant,  may  be 
considered  as  satisfactory.    Gordon  s  formula  very  fairly  satisfies  this  requirement,  being  of 

the  form  /  =  ^  1  /\^'  convenient  of  application,  however,  as  the  one  now 


iroposed,  which  is  of  the  form  p  =  f  —  b\  —  \  .    If  the  coefficient  d  in  this  formula  be  evaluated 


so  as  to  make  the  locus  tangent  to  that  of  Euler's  curve,  and  the  formula  used  for  all  lengths 
up  to  this  point  of  tangency,  it  will  give  values  as  near  the  average  of  those  obtained  from 
actual  experiments  as  possible. 

To  find  the  Equation  of  the  Parabola  having  its  Vertex  at  the  Elastic  Limit  on  the  Axis  of 
Loads,  and  Tangent  to  Eider's  Curve. 

For  hinged  and  for  flat  ends  an  empirical  coefficient  must  be  found  which  will  make 
Euler's  curve  best  fit  the  observed  strength  of  very  long  columns.  The  general  form  of 
Euler's  formula,  for  all  varieties  of  end  conditions,  is 


P 


en 


'E 


(12) 


For  hinged  ends  we  shall  use 


16, 


and  for  square  or  flat  ends  we  will  make 


25. 


The  value  of  E  will  be  taken  as  28,500,000  for  steel  and  27,000,000  for  wrought-iron. 


*  See  Mr.  Marshall's  paper  referred  to  in  foot-note,  p.  143. 


COLUMN  FORMULA.  149 
We  have,  therefore,  as  Euler's  formula  for  Wrought-iron  Compression  Members, 


For  Hinged  Ends,    p  ~ 


432,000,000 


(13) 


For  Flat  Ends, 


675,000,000 


These  are  the  curves  BCD  and  FGH  in  Fig.  207.  The  line  ABF  marks  the  elastic 
limit  of  wrought-iron,  and  A'B'F'  the  elastic  limit  of  steel.  The  tangent  curves  AC  and 
AG  are  parabolas,  with  vertex  at  A  and  an  axis  in  AO,  drawn  tangent  to  Euler's  curve  for 


hinged  and  for  flat  ends  at  the  points  C  and  G,  respectively.  These  are  the  loci  of  the 
wrought-iron  formulae  for  ordinary  lengths,  while  the  analogous  curves  A'C  and  A'G'  are  the 
loci  of  the  formulae  for  steel  columns.  The  complete  loci  showing  the  strength  for  all  lengths 
are  ACD  and  AGH  for  wrought-iron,  and  A' CD  and  A'G'H  for  steel  columns.  The  equa- 
tion of  these  tangent  parabolas  takes  this  form  : 


Tangent  Parabola,    y  z=  f  —  bx^. 
The  equation  of  Euler's  formula  takes  the  form, 


(15) 


Euler's  Curve,    y  —   ...(16) 

We  wish  now  to  find  the  value  of  x,  ^=  -j,  at  the  point  of  tangency,  and  the  value  of  the 
coefficient,  b,  which  will  pass  the  parabola  through  this  point.    The  equations  of  condition  are 


ISO 


MODERN  FRAMED  STRUCTURES. 


dy 

found  by  making  ^  equal  for  the  two  curves,  and  then  equating  the  two  values  of  y,  or 
making  the  equations  simultaneous  for  the  point  of  tangency.  Thus, 


and 


dy 

^  from  eq.  (15)  r=  —  2bx 


dy  2k 
^-from  eq.(i6)=--^ 


*.  X  — 


(17) 


Making  the  two  values  oi  y  equal  for  the  point  of  tangency,  we  have 


X-  ■  X 
From  (17)  and  (18),  by  elimination,  we  may  find 

'Tk 


(18) 


/* 

—    and    b  = 
f  Ak 


Eq.  Elder's  Curve,  p 


(19) 
(20) 


which  substituted  in  (15)  and  (16)  we  obtain  : 

PI  A* 

Eq.  Tang.  Parabola,    p  =  f  —  ^\  /  •  ..••«. 

k 

For  wrought-iron,  hinged  ends,   k  =z  16E  =  432,000,000. 

For  steel,  hinged  ends,  k  =  16E  =  456,000,000. 

For  wrought-iron,  flat  ends,        k  =  2^E  =  675,000,000. 

For  steel,  flat  ends,  k  =  2^E  =  7  12,000,000. 

For  wrought-iron,  the  elastic  limit,  —  /  =  34.O0O. 

For  mild  steel,  the  elastic  limit,  —  f  —  42,000. 

The  elastic  limit  of  rolled  metals  increases  with  the  amount  of  work  put  on  the  bar,  or  it 
varies  inversely  as  the  thickness  of  the  finished  plate  and  inversely  with  the  temperature 
when  leaving  the  rolls.  Since  compression  members  are  made  up  from  sections  having  thin 
webs,  from  f  to  f  inch  thickness,  the  average  elastic  limits  of  wrought-iron  and  mild  steel 
will  be  about  as  here  taken.*    Hence  we  may  write  the  following  numerical  formulae, 

I  I  /2k 

remembering  that  the  parabolic  law  only  applies  from  —  =  o  to  -  =  a  /       as  shown  above 

r  I'        y  f 

in  the  value  found  for  x  at  the  point  of  tangency.    With  the  values  of  k  and  x  here  taken,  we 


have 


FORMULA  FOR  THE  ULTIMATE  STRENGTH  OF  COLUMNS. 

-  - 170,  / = 34,000  -  .671  , 


r  I 


For  Wrought-iron  Columns,  Pin  Ends,      -{  I  —  432,000,000 


(21) 


*  The  great  change  in  the  elastic  limit  for  different  thicknesses  of  finished  sections,  and  for  different  conditions  ot 
rolling,  especially  the  temperature  when  leaving  the  rolls,  largely  accounts  for  the  extraordinary  range  of  the  results 
of  the  experimental  tests  of  wrought-iron  and  steel  columns. 


COLUMN  FORMULA. 


For  Wrought-iron  Columns,  Flat  Ends,     ^  /  _ 


2IO,  p 


2IO,  p 


34,000  — .43  (^) 
675,000,000 


(22: 


For  Mild  Steel  Columns,  Pin  Ends, 


-  1 50,   /  =  42,000  —  .97 


For  Mild  Steel  Columns,  Flat  Ends, 


I  _ 

^>i5o,  P 


-  /  _ 
/  -  ^  190, 


90,  p 


456,000,000 

42,000  -  .62  (^) , 
712,000,000 


(23) 


(^) 


(24) 


In  actual  practice  -  is  nearly  always  less  than  I  50,  and  usually  less  than  100,  so  that  the 
formula,  /  =/—  covers  the  entire  range  of  ordinary  practice. 


For  Cast-iron,  Round  Ends, 


-  <  70,   /  =  60,000  -  -(-j  , 


K  /  _  144,000,000 
1_-  70,   /  =  


For  Cast-iron,  Flat  Ends, 

For  White  Pine,  Flat  Ends, 

For  Short -leaf  Yelloxv  Pine,  Flat  Ends, 

For  Long-leaf  Yellow  Pine,  Flat  Ends, 
For  White  Oak,  Flat  Ends, 


I  _ 

r  < 

120. 

I  ^ 
r  > 

120, 

/  _ 
d  < 

60. 

/  _ 

60, 

/  _ 
d< 

60, 

I  _ 
a  < 

60, 

400,000,000 


(25) 


(26) 


2500 
3300 
4000 

3500 


o4r 


(27) 


In  Fig.  208  are  shown  the  plotted  results  of  all  the  most  reliable  tests  of  full-sized  columns 
ever  made.  The  loci  of  the  Parabolic  Formulae  given  above,  of  the  Gorden-Rankine  Formulae, 
and  of  Johnson's  Straight-line  Formulae  are  all  drawn  for  both  iron  and  steel. 

138.  Johnson's  Straight-line  Formulae. — In  1886  Mr.  Thos.  H.  Johnson,  M.  Am. 
Soc.  C.E.,  presented  a  paper  to  the  American  Society  of  Civil  Engineers  in  which  he  showed 
that  a  straight-line  formula  could  be  made  to  fit  the  plotted  observations  of  column  tests,  as 
well  as  any  curve,  for  all  the  ordinary  lengths.    He  used  Euler's  curves  for  the  great  lengths 

*  In  these  formulae  for  wooden  columns  is  the  least  lateral  dimension.  The  numerical  coefficients  are  based  on 
Prof.  Lanza's  experiments.    See  Lanza's  Applied  Mechanics. 


Modern  framed  structures. 


to  w  hich  it  is  applicable,  and  made  his  straight-line  loci  tangent  to  these  curves.  Although 
these  linear  equations  can  be  made  to  fit  the  observed  results  for  ordinary  lengths  as  well  as 
any  other  (see  these  loci  in  Fig.  208),  they  give  too  great  values  of  the  ultimate  strength  for 


Bank  Not.  Co.N.Y. 


Fig.  208. 


the  shorter  lengths,  provided  failure  be  taken  at  the  elastic  limit  of  the  material,  or  where  the 
member  takes  on  an  appreciable  permanent  set.  Mr.  Johnson's  formulae  are,  however,  the 
simplest  ever  yet  proposed  and  have  come  into  very  general  use.    They  are  as  follows  : 

/ 

Wrought-iron — Hinged  Ends,  p  —  42,000  —  157  -. 

/ 

«  Flat  "       p  =  42,000  —  128  -. 

r 

I 

Mild  Steel,       Hinged  Ends,   p  =  52,500  —  220  -. 


I<  u 


I 

Flat  "  =  52,500  —  179 -. 


COLUMN  FORMULAE. 


153 


139.  Formulae  to  be  Used  in  Dimensioning.* — All  working  formulae  must  include  a 
factor  of  safety.  It  has  been  customary  to  introduce  this  as  a  common  factor  in  the  right-hand 
member  of  the  ultimate  strength  formula.    This  is  clearly  wrong.    Referring  back  to  for- 


m 


ula  (i),  p.  144,  we  see  that  the  second  term  in  the  denominator,  — ',  represents  not  only 

any  known  or  unknown  eccentricity  of  loading,  but  also  all  structural  weaknesses,  like  initial 
bending,  internal  stress,  unsymmetrical  cutting  away  of  parts,  etc.,  and  hence  some  value 
must  be  given  to  this  term  even  when  the  loads  are  assumed  to  be  symmetrically  placed. 
The  giving  to  this  term  an  arbitrary  value  is  therefore  equivalent  to  introducing  a.  factor  of 
safety.  If,  for  centrally  placed  loads,  we  give  this  term  a  value  of  unity,  this  is  equivalent  to 
making  the  factor  of  safety  two  for  short  columns,  since  the  third  term  in  this  denominator 
practically  vanishes  for  short  lengths.  This  factor  may  also  be  regarded  as  a  factor  of  safety, 
for  the  stress  on  the  most  compressed  fibre,  or  as  a  factor  of  safety  as  to  the  maxiviiivi  stress. 
If  now  we  arbitrarily  introduce  a  factor  of  safety  of  two  as  common  to  the  entire  denomi- 
nator, we  may  call  tliis  a  factor  of  safety  as  to  the  loads,  which  also  applies  to  the  stresses  as 


l(l,()00 


7,500 


5,000 


;,500 


LOCI  OF  COLUMN  FORMULAE  FOR 
PIVOTED  AXD  FOR  PIN  ENDS. 

(A)  — Crehore's  theoretical  form  of  Ranlyne's  Formula  for 

Pivoted  Ends:  i'^Ki+^rj  (4)') 

(B)  ~Tlie  Author's  Parabolic  Formula  for  Pivoted  Ends: 

p=i(/-1.53(^f) 

(C)  — The  Author's  New  Formula  for  Pivoted  Ends: 

(D)  -  The  Author's  Parabolic  Formula  for  Pin  Ends: 

p=i(/-0.97  (if.)') 

(E)  ~  The  ordinary  form  of  Rankine's  Formula  for  Pin  EnJsr 

(F )  — The  Author's  New  Formula  for  Pin  Ends: 


NOTATION: 
p  =  working  load  per  sq.  in. 
/=  apparent  Elastic  limit 

=  42,000  for  mild  steel. 

=  modulus  of  Elasticity 

=  28,500,000. 
/  —length  of  column, 
r  =least  radius  of  gyration 


:;o 


40 


8P 


100 


120 


140  KiO 

Fig.  2o8«. 


180 


200 


220 


2(10 


2t<0 


a  matter  of  necessity.    The  result  is  we  now  have  a  factor  of  safety  of  two  for  very  long 


vy. 

columns  (where  the  terms  i  -(-  ~  are  of  little  relative  value),  and  of  four  for  very  short 


r 

7^ 


columns  (where  the  last  term  in  '^--j  has  little  relative  value).     This  is  as  it  should  be,  since 

the  strength  of  a  very  long  column  is  not  a  matter  of  stress  on  the  most  compressed  fibre, 
but  purely  a  matter  of  stiffness,  or  ability  to  remain  straight,  or  unbent,  under  its  load. 
Here,  therefore,  we  need  only  a  factor  of  safety  as  to  the  load.  By  proceeding  in  this  man- 
ner we  may  say : 

(1)  The  column  will  stand  even  though,  for  unknown  reasons,  the  stress  on  the  most 
compressed  fibre  is  twice  as  much  as  it  has  been  computed;  and 

(2)  It  will  also  stand  even  though  the  load  is  twice  as  great  as  it  has  been  assumed. 
When  a  factor  of  safety  of  four  is  taken  as  a  common  factor  for  the  entire  denominator, 

we  are  making  this  wholly  a  load  factor  for  very  long  columns,  which  is  unreasonable,  as  we 
do  not  propose  to  provide  for  four  times  as  great  a  load  as  has  been  a.ssumed.  Making  these 
numerical  changes,  equation  (i),  p.  144,  becomes  equation  (28)  below. 

*  This  article  added  in  the  Sixth  Edition. 


12,500 


10,000 


MODERN  FRAMED  STRUCTURES. 


2,500 


7,5()0  A 


5,000 


1110  T-T)  150  175 

Fig.  TO&b.    Equation  (28). 


12,500 


110,000 


7,5«0 


5.000 


2,500 


100  l:.'r)  150  175 

Fig.  2o8i-.    Equation  (29). 


12,500 


10,000 


7,500 


6,000 


2,500 


100  1'.'5  151)  175 

Fig.  20§(/.    (Equation  30). 


3U0 


COLUMN  FORMULA. 


The  locus  of  this  equation  is  curve  C  in  Fig.  2o8fl.  It  should  be  compared  with  the 
author's  parabolic  formula  as  described  on  the  previous  pages,  here  shown  as  curve  B,  and 
also  with  Crehore's  theoretical  form  of  Rankine's  formula,  here  shown  in  curve  A.  All  these 
formulae  and  loci  are  for  columns  with  pivoted  ends. 

For  pin  ends,  \oE  becomes  \GE,  by  experiment,  as  described  at  the  bottom  of  p.  148, 
and  for  square  ends  it  becomes  2^E.  Making  this  change  for  pin  ends,  we  obtain  the  curve 
marked  F  \x\  Fig.  2oZa,  which  should  be  compared  with  the  author's  parabolic  locus,  curve  D, 
and  with  the  ordinary  locus  of  Rankine's  equation  for  pin  ends  as  there  given.  A  study  of 
these  curves  reveals  the  following  relations  : 

For  Pivoted  Ends  (comparing  with  curve  C). 

(1)  The  Crehore  locus  (curve  is  much  too  low,  as  shown  by  the  experimental  results 
given  in  Fig.  208.* 

(2)  The  author's  parabolic  formula  (curve  B),  having  a  factor  of  safety  of  four  throughout, 
runs  too  low  for  long  columns,  and  a  little  too  high  for  the  ordinary  lengths.  This  is  also  the 
case  to  a  similar  degree  with  the  formulas  for  pin  ends  (curve  D),  and  it  is  also  the  case  for 
square  ends,  not  shown  in  the  diagram. 

For  Pill  Ends  (comparing  with  curve 

(3)  The  locus  of  the  ordinary  Rankine  formula  lies  well  below  that  of  the  author's  new 

formula     until  a  value  of  —  =  270  is  reached,  where  they  intersect.   This  difference  is  nearly 

20  per  cent  for  the  ordinary  lengths.  A  similar  difference  would  appear  for  the  formulae  for 
flat  ends  if  their  loci  were  plotted. 

Taking,  then,  the  formulae  C  and  F,  and  an  analogous  one  for  flat  ends,  we  have  the 
following 

WORKING  FORMULA  AND  DIAGRAMS. 

^  ^  (28) 

\oE^^  r 

For  Pin  Ends  (Fig.  208c),         p  =  ^/  T^TWi  i^-^  \  (29) 


For  Pivoted  Ends  (Fig,  2o8(5),    p  —  -  /  — :  -r-\ 


For  Flat  Ends  (Fig.  2oM),       p  =  V  7~^ivi\  (3°) 

2^E  \  rl 

Giving  to  f,  the  "  apparent  elastic  limit,"  values  from  30,000  to  50,000  for  different  grades  of 
wrought  iron  and  steel  {/  =  34,000  for  wrought  iron  and  /  =  42000  for  mild  steel  being  safe 
reasonable  values  for  shape  irons  in  thin  sections,f  and  taking  E  =  28,500,000  in  all  cases, 
there  results  the  various  curves  in  these  figures.    In  finding  unit  stresses  to  use  in  designing, 

/ 

therefore,  it  is  only  necessary  to  know  the  value  of  -  for  the  given  column,  and  the  kind  of 

metal  (/)  employed.  Since  r  does  not  change  appreciabl)'  for  different  weights  of  section, 
but  only  for  different  sizes,  when  the  lateral  dimensions  of  the  column  are  given,  the  maker's 
handbooks,  or  Osborn's  Tables  of  Radii  of  Gyration,  will  give  at  once  the  value  of  r  with 
sufficient  exactness.    The  working  stress  (/>)  can  then  be  taken  from  the  diagrams. 

Note.— For  short  lieavy  columns,  such  as  are  used  in  buildings,  the  bending  of  the  columns  may  be 
neglected  and  provision  may  be  made  for  eccentric  loads,  as  indicated  on  p.  453  in  the  formula 

A  =  pr  +  ^kP,)  (31) 

where  A  =  area  of  cross-section  of  columns; 

p  =  maximum  crushing  stress  to  be  allowed  in  practice; 
P  =  total  load  on  the  column  ; 
Pe  =  eccentric  load  ; 

i  =  ratio  of  eccentricity  of  Pg  to  the  half  width  of  the  column  in  that  direction. 

*  This  has  been  pointed  out  by  the  author  as  being  true  on  both  theoretical  and  experimental  grounds.  See  Engi- 
neering News,  vol.  xxjcvjl  (May  20,  1897),  p.  311,  f  See  Johnson's  Materials  of  Construction, 


154 


MODERN  FRAMED  STRUCTURES. 


CHAPTER  X. 

COMBINED  DIRECT  AND  BENDING  STRESSES.    SECONDARY  STRESSES. 

140.  Examples  of  Combined  Stresses. — Whenever  a  tension  or  compression  member 
occupies  a  horizontal  position  it  is  evidently  subjected  to  a  cross-bending  stress  from  its  own 
weight.  If  other  transverse  external  forces  come  upon  it,  it  is  still  further  stressed  in  this 
way.  In  case  the  direct  loading  (tension  or  compression)  be  not  centrally  placed  over  the 
centre  of  gravity  of  the  cross-section,  or  if  the  member  itself  be  bent  from  a  right  line,  then 
there  would  be  a  bending  moment  upon  the  member  equal  to  the  direct  loading  into  its  arm, 
or  the  deviation  of  the  axial  line  of  the  column  from  the  right  line  joining  the  centres  of 
gravity  of  the  end  sections.  In  all  such  cases  the  member  should  be  dimensioned  for  both 
direct  and  cross-bending  stress,  and  the  computation  made  on  the  assumption  that  both  kinds 
of  loads  act  simultaneously. 

It  is  always  poor  economy  to  subject  a  member  to  bending  stress.  It  must  be  dimen- 
sioned for  the  stress  on  the  extreme  fibres,  while  the  strength  of  the  interior  portion  of  the 
section  remains  unused.  A  framed  structure  should  always  be  designed  so  as  to  obtain  only 
direct  stresses  in  all  its  members,  and  then  the  entire  cross-section  is  available  to  resist  the 
force  coming  upon  it.  It  is  for  this  reason  that  all  joints  should  be  designed  to  bring  the  cen- 
tre of  gravity  lines  of  all  the  members  to  a  common  point,  so  as  to  avoid  secondary  stresses 
from  bending.  The  use  of  knee,  portal,  and  sway  bracing  also  should  be  avoided  whenever 
possible,  as  it  usually  is  in  buildings,  trestles,  towers,  roofs,  and  deck  bridges. 

141.  Action  of  Direct  and  Bending  Stresses. — The  effect  of  any  finite  cross-bending 
load  upon  any  member  is  to  produce  a  corresponding  deflection  in  it.  The  effect  of  the 
direct  loading  is  to  increase  the  bending  moment,  and  therefore  the  deflection,  if  acting  so  as 
to  compress  the  member,  while  the  reverse  is  true  for  an  extending  force.  When  the  deflec- 
tion becomes  appreciable,  the  efYect  of  these  direct  forces  upon  the  cross-bending  stresses  is  too 
great  to  be  neglected.  It  is  common  to  assume  that  these  two  classes  of  external  forces  act 
independently,  and  to  compute  their  separate  effects,  add  the  resulting  stresses  together,  and 
to  dimension  the  member  accordingly.  In  the  following  analysis  they  are  treated  as  acting 
simultaneously  and  the  true  cross-bending  stresses  found. 

142.  Derivation  of  a  General  Formula  for  Combined  Stresses. 

Let        —  bending  moment  at  point  of  maximum  deflection,  from  cross-bending  external 
forces  and  from  eccentricity  of  position  of  longitudinal  loading  ; 
7/,  =  maximum  deflection  of  member  from  all  causes  acting  simultaneously; 

—  bending  moment  from  the  direct  loading,  P,  into  its  arm,  z\  ,  =  Pv^ ; 
P  =  total  direct  loading  on  member,  tension  or  compression  ; 

=  unit  stress  on  extreme  fibre  from  bending  alone  at  section  of  maximum  bend- 
ing moment,  or  of  maximum  deflection,  as  the  case  may  be,  in  pounds  per 
square  inch ; 
/  =  length  of  member  ; 
^,  =  distance  from  centre  of  gravity  axis  to  the  extreme  fibre  under  consideration 

on  which  the  stress  from  bending  is  ; 
£  =  modulus  of  elasticity  : 


COMBINED  DIRECT  AND  BENDING  STRESSES. 


155 


/  =  moment  of  inertia  of  the  cross-section; 
b  —  breadth  of  a  solid  rectangular  section ; 

h  —  height  of  section,  out  to  out,  in  the  plane  in  which  bending  occurs,  =  2y^  for 
symmetrical  sections. 
=  unit  stress  in  member  from  the  direct  loading,  supposed  to  be  uniformly 

distributed,  = 

A 

f  —  total  maximum  unit  stress  on  extreme  fibre,  =  -\-fi' 
AU  dimensions  in  inches,  and  forces  in  pounds. 

In  all  cases  of  deflection  of  beams  of  constant  moments  of  inertia  the  maximum  deflec- 
tion, in  terms  of  the  stress  on  the  extreme  fibre,  is  given  by  the  equation 

where     is  a  numerical  factor. 

Thus  for  the  extreme  cases  of  a  beam  supported  at  the  ends  and  loaded  at  the  centre,  and 
for  the  same  beam  loaded  uniformly,  the  value  of  k  is  -^-^  in  the  former  case  and  -^-^  in  the 
latter.* 

Since  almost  all  cases  in  practice  correspond  more  closely  with  the  condition  of  uniform 
loading,  the  value  of  k  will  be  taken  as  -f^  or  yi„.  We  then  have  as  the  general  relation 
between  the  deflection  of  a  beam  and  the  stress  on  its  extreme  fibre 

^'=7^,  

In  all  cases  now  under  consideration  there  are  the  two  bending  moments  acting  on  the 
member,  and  J/,,  which  may  be  of  the  same  or  of  opposite  signs.  The  moment  of  resist- 
ance, or  the  moment  of  the  direct  stresses,  developed  at  any  section  must  be  equal  to  the 
algebraic  sum  of  the  moments  from  the  external  forces,  and  hence  we  may  write 

M,=  ^-:^M,±M,  =  M,±Pv,,  (3) 

the  positive  sign  to  be  used  for  members  under  compression,  and  the  negative  sign  for  mem- 
bers under  tension. 

Putting  for     its  value  from  (2),  we  have 

/i  —        pp  '   ^4; 

loE 

where  the  negative  sign  is  to  be  used  for  compression  members,  and  the  positive  sign  for 
tension  members. 

This  formula  is  perfectly  general,  and  applies  rigidly  to  all  forms  of  section  and  to  all 
forms  of  loading  without  material  error.  The  second  term  in  the  denominator  takes  account 
of  the  bending  moment  ■=  Pi\ ,  and  can  be  neglected  in  all  cases  where  the  amount  of  the 
bending  is  known  to  be  inappreciable. 

Case  I.  Tension  and  Cross-bending. 

143.  Example  i.  Find  the  stress  in  the  extretne  tower  fibres  of  an  eye-bar  ivhen  used  in  a  horizontal 
position  and  subjected  to  its  07vn  weight. 

Here  the  transverse  moment  is  that  due  to  its  own  weight.    If  we  take  a  section  at  the  centre  of  the 

*  These  factors  are  given  in  column  four  of  the  table  on  pages  132  and  133,  except  that  in  that  table  ti  is  used  where 

here^i  is  employed,  which  is  equal  to  -;  hence  the  factors  here  used  are  one-half  of  those  of  the  table. 

2 

f  This  is  for  a  member  free  to  turn  at  the  ends.  For  a  member  fixed  at  the  ends  use  -^^  or  0.031,  and  for  one  end 
fixed  use  ./j  or  0.043  for 


MODERN  FRAMED  STRUCTURES. 


bar  and  equate  the  moment  of  resistance  of  the  internal  stresses  with  the  algebraic  sum  of  the  moments  due 
to  the  weight  of  the  bar  and  to  the  pull  upon  it  into  the  deflection  at  the  centre,  which  is  the  arm  of  this 
force,  we  have 

or 

where  w  =  weight  of  bar  per  inch  =  o.zSM. 

Putting  Vi  =  — —  from  (2),  and  P  =  Jtbh,  and  taking  E  =  28,000,000,  we  have,  from  (5), 

4,800,000^ 

/•  =  7777  (6) 


fi  +  23,000,0001 -J  j 


as  the  tensile  stress  per  square  inch  in  the  bottom  fibres.    The  total  stress  in  these  extreme  fibres  is  therefore 

4,800,000^  P 
/=/.+/.  =  TTr.  +  JJ,  (7) 


Iky 

fl  +  23,000,0001  yj 


144.  Effect  of  Height  of  Bar  on  Fibre  Stress. —  From  eq.  (7)  it  is  seen  that  the  stress  on  the  extreme  fibre 
from  bending  of  an  eye-bar  under  its  own  weight  is  a  function  of  A,  fi,  and  /.  If  fi  and  /  be  considered 
constant  and  h  allowed  to  vary,  we  may  find  the  depth  of  bar  giving  maximum  fibre  stresses  by  differentiat- 
ing eq.  (6)  with  reference  to  f\  and  h,  placing  the  first  differential  coefficient  equal  to  zero,  and  solving  for  k. 
This  gives 

o  =  23,000,000//-  —  f-il. 


dh 


or 


'=i^^f^   

for  height  of  bar  giving  maximum  fibre  stresses  from  their  own  weight. 

If  this  value  of  h  be  substituted  in  eq.  (6),  we  have,  as  the  stress  on  the  extreme  fibres  from  bending 
under  their  own  weight,  for  the  depths  giving  maximum  stresses, 

500/ 

fl{max.)  =  —J=r  (Q) 

The  following  table  gives  the  depths  of  bars  having  maximum  fibre  stresses  from  bending  under  their  own 
weights,  and  also  the  amounts  of  these  stresses,  for  different  working  tensile  stresses  in  the  bar  and  for 
different  panel  lengths. 

TABLE  OF  DEPTHS  AND  MAXIMUM  FIBRE  STRESSES. 


Working 
Tensile 
Stresses 
in  pounds 
per 
square 
inch. 


8000 
1 0000 
12000 
14000 
16000 


Length  of  Eye-bars  in  Feet. 


15 


Depth. 


In. 

3 

3 

4 

4 

4 


Fibre 
Stress. 


Lbs., 
Sq.  In. 

lOIO 
900 
820 
760 
720 


20 


Depth. 


In. 

4-  5 

5-  1 
5-6 
6.0 
6.4 


Fibre 
Stress. 


Lbs., 
5q.  In. 
1340 
1200 
1 100 
1020 
960 


25 


Depth. 


In. 


Fibre 
Stress. 


Lbs., 
Sq.  In. 
1680 
1500 

1370 
1270 
1200 


30 


Depth. 


In. 

6.8 

7.6 

8.4 
9.0 
9.6 


Fibre 
Stress. 


Lbs., 
Sq.  In. 
2020 
1800 
1640 
1520 
1440 


35 


.^Depth. 


In. 


Fibre 
Stress. 


Lbs., 
Sq.  In. 
2350 
2100 
1920 
1780 
1680 


40 


Depth. 


In. 

9.1 
10. 1 
II .  I 
12.0 
12.9 


Fibre 
Stress. 


Lbs.. 
Sq.  In. 
2690 
24>DO 
2190 
2030 
1920 


For  any  given  case,  where  the  depth,  length,  an(|  total  pull  per  square  inch  are  given,  use  eq.  (6)  for 
finding  fibre  stress  from  bending. 


COMBINED  DIRECT  AND  BENDING  STRESSES. 


157 


145.  Example  2.  Find  the  stress  in  the  extreme  fibres  of  an  eye-bar  (or  any  rectangular  tension  member) 
vjhicJL  also  carries  an  external  load  of  Wi  lbs.  per  linear  inch. 

This  problem  is  exactly  similar  to  the  former  one,  except  that  {w  +  Wj)  is  now  to  be  used  where  w  alone 
was  used  before.    Using  this  notation,  we  have,  analogous  to  (5), 


from  which  we  have,  as  before. 


^^1±^_P.,^J^  00) 

o  0 


io5,ooo(w  +  w,) 


db-^^  +  \±0,QOQb{^^ 
1000  \l I 


Thus  if  the  cross-ties  of  a  bridge  should  rest  directly  on  the  eye-bars,  luj would  be  one-half  of  a  total 
panel  load,  and  the  value  of  w  would  be  relatively  insignificant. 

If «/  +  were  taken  as  900  lbs.  per  foot,  or  75  lbs.  per  inch,  and  two  eye-bars  in  each  chord  with  a 
section  of  4  in.  x  i  in.  be  used,  and  there  be  a  pull  of  14,000  lbs.  per  square  inch  on  these  bars,  and  they  be 
assumed  to  be  15  ft.  long,  we  have 

w -f      =  75  ;    b  =  2\  /2=  14,000;    /^  =  4;    and  /=i8o, 

whence  fi  =  25,600  lbs.  per  square  inch  from  bending,  or  f  =fi  +  f^i^  39,600  lbs.  f)cr  square  i}ich  total  stress 
on  the  extreme  lower  fibres.  This  being  beyond  the  elastic  limit,  the  bars  would  permanently  elongate  on 
that  side  and  then  be  straightened  again  when  the  load  passed  off,  and  in  this  way  the  bars  would  become 
"fatigued,"  and  finally  would  fail. 

It  may  be  interesting  to  note  that  if  the  pull  in  these  bars  be  neglected  in  the  computation  of  bending 
moment,  and  the  same  transverse  load  applied,  the  fibre  stress  would  be  about  ^"j ,000  lbs .  per  square  inch 
from  cross-bending  alone.  The  necessity  of  taking  account  of  the  simultaneous  action  of  the  two  moments 
is  thus  shown. 

146.   Example  3.  What  is  the  stress  from  flexure  in  a  lateral  tie-rod  i  in.  square,  40  ft.  long,  hanging 
freely  from  end  supports,  and  strained  up  with  an  initial  pull  of  10,000  lbs..' 
From  eq.  (4)  we  have 

.        ^1/1  8000  X  i  4000  . 

/i  =  —  =  =   =  480  lbs.  per  square  mch. 

I       10,000  X  480  X  480     0.08  4-  8.23  V  ^ 

I  -I   —  H  —  ^ 

loE      12  280,000,000 

The  deflection  is  found  from  the  formula 

Vi  —  —  .         =  0.82  in. 

24  Eh 

This  member  is  so  shallow  as  to  act  like  a  wire,  the  depth  giving  maximum  fibre  stress  for  this  span  and 
unit  stress  being,  from  eq.  (8),  equal  to  10  in. 


Case  II.  Compression  and  Cross-bending. 

147.  Example  4.  Find  the  extreme  fibre  stress  in  cross-bending  arising  from  its  own  weight  and  its 
compressive  load,  of  a  top-chord  section,  25  ft.  long,  composed  of  two  15-inch  channels  of  120  lbs.  per 
yard  and  one  20  in.  x  f  in.  top  plate. 

For  this  case  the  general  equation  (4)  takes  the  form 

/'=   —  •  02) 

The  endwise  loading  is  supposed  to  come  upon  the  section  at  the  centre  of  gravity  axis.  The  maximum 
fibre  stress  will  come  on  the  top  plate,  which  is  the  most  compressed  side  of  the  section.  The  value  of  /  for 
this  section  is  1155,  and  of  y^  for  the  plate  side  6.04  in.    Taking  E  =  28,000,000  and  /j  =  7000  lbs.,  P  is 


MODERN  FRAMED  STRUCTURES. 


31^  X  7000  =  220,000  lbs.  The  bending  moment  at  the  centre  from  its  own  weight  is  98,400  in.-lbs.  The 
length  is  300  inches.    We  have,  therefore, 

i^/iVi  98,400  X  6.04 

/i  =  -  =  ^— ^  =  550  lbs. 

•'  PP  220,000  X  90,000 

loE      ^'^^  280,000,000 

This  is  the  stress  on  the  plate  from  cross-bending  only. 

148.  Example  5.  What  is  the  effect  of  loading  the  above  column  at  the  centre  line  of  the  channel-bars 
composing  its  sides  ? 

This  is  a  common  error  in  construction  and  deserves  careful  consideration.  The  centre  of  gravity  axis 
of  this  section  is  1.83  in.  from  the  centre  line  of  the  channels.  If  we  now  omit  from  consideration  the 
weight  of  the  member.  Mi  will  be  composed  wholly  of  the  longitudinal  force  A  into  the  eccentricity  of  the 
loading,  which  is  1.83  in.    Hence  Mi  =  220,000  x  1.83  =402,600  in.-lbs. 

In  this  case  the  plate  side  will  be  convex,  and  the  maximum  compressive  stress  will  be  found  on  the 
latticed  side  of  the  member.    For  this  side/i  =  94  in.    The  other  factors  are  the  same  as  before.  Therefore 

M,yi        402,600  X  94        „  .  . 

/»  =  ^  =    ii^^^ji  —  3470  lbs.  per  square  inch. 

^  loE 

149.  Example  6.   What  is  the  combined  effect  of  weight  of  member  and  eccentric  loading  f 

This  is  a  combination  of  the  conditions  named  in  Examples  3  and  4,  this  combination  being  a  common 
practice  for  upper  chord  members.    In  this  case  the  algebraic  sum  of  these  two  effects  must  be  taken.  Thus 
the  weight  of  the  member  would  produce  a  compression  of  550  lbs.  per  square  inch  in  the  upper  fibres  and 
9-33 

a  tension  of  y —  x  550=  850  lbs.  on  the  lower  fibres.    The  eccentric  loading  gave  3470  lbs.  compression  on 
6.04 

these  latter,  the  algebraic  sum  of  the  two  being  2620  lbs.  per  square  inch  compression. 

To  this  must 
Making  a  total  of 


P  220,000 

To  this  must  be  added  the  uniformly  distributed  load  y2  =  — -  =        ^    =  7000  lbs.  per  square  inch. 


f =ft  +  /a  =  2620  +  7000  =  9620  lbs.  per  square  inch. 

The  total  compression  in  this  member  is  about  37I  per  cent  greater  than  the  ordinary  specification 
would  allow. 

150.    Example  7.  Compute  the  jnaximian  stress  on  the  extreme  fibre  of  a  top-chord  section  which  is  used  to 
carry  the  cross-ties  on  a  railway  deck-bridge. 
Let  /  =  20  ft.  =  240  in. ; 

w  =  dead  load  per  foot  per  truss  =  500  lbs. ; 
p  =  live  load  per  foot  per  truss  =  3400  lbs.  ; 
P  =  compressive  stress  in  chord  =  400,000  lbs. ; 
Ml  =  moment  from  transverse  load  =  2,340,000  in.-lbs. 
Let  the  chord  be  made  up  of 

One  top  plate,         24  in.  x  1  in.,  =  21.0  sq.  in. 

Two  top  angles,  4  in.  x  4  in. — 42  lbs.,  =  8.4  " 
Two  side  plates,      24  in.  x  ^  in.,  =  39.0  " 

Two  bottom  angles,  6  in.  x  4  in. — 70  lbs.,  =  14.0  " 

Total  area  of  section  =  82.4  sq.  in. 

The  moment  of  inertia  of  this  section  is  7436,  and  the  neutral  axis  lies  10.58  in.  from  the  upper  side  of 
the  chord. 

From  eq.  (4)  we  have 

Miy<               2,340,000  X  10.58       •  .  , 

/,  =  •   =  ^  =  3370  lbs.  per  square  men. 

PI'  400,000  X  400  X  144 

/  —  •     7436  —  ■ — ■ 

loE  280,000,000 

For  this  section  and  compressive  load  we  have 

P  400,000 

/a  =  —  =  ~  =  4850  lbs.  per  square  inch, 

W  02.4 


COMBINED  DIRECT  AND  BENDING  STRESSES. 


159 


The  total  compressive  stress  on  the  extreme  fibres  at  top  is  therefore 

f  —  fi    fi  =  3370  +  4850  =  8220  lbs.  per  square  inch. 

The  second  term  in  the  denominator  of  formula  (4)  represents  the  effect  of  the  direct  compressive 
stress  acting  with  the  arm  Vi ,  the  deflection  of  tlie  member.  In  this  case,  where  the  member  is  very  rigid 
and  deflects  very  little,  the  effect  is  very  small,  this  term  being  but  82,  whereas  /  =  7436.  The  effect  of 
neglecting  this  term  in  this  case,  therefore,  would  be  to  give  a  fibre  stress  i-jV  per  cent  too  small.  For  more 
flexible  members  the  effect  of  neglecting  this  term  is  greater. 

151.  Fixed  End  Posts  with  the  Upper  Ends  not  Fixed  in  Direction. — Let  Fig.  208^: 
represent  a  portal,  or  a  bent  of  an  elevated  railroad  or  of  a  steel  frame  building,  having  the 

posts  fixed  in  direction  at  the  ground  and  having  a  single 
system  of  diagonal  bracing  as  shown.  The  problem  is  to  find 
the  point  of  inflection  from  the  base,  and  then  the  values 
of  the  reactions  //"and  Fupon  the  columns.  These  reactions 
will  be  the  same  as  in  form  i,  Art.  115,  Chap.  VII,  when  the 
point  of  inflection  is  treated  as  the  base  of  the  column.  The 
problem  is  simplified  by  considering  a  simple  beam,  as  in  Fig. 
20^h,  fixed  at  one  end  and  subjected  to  the  forces  R  and  Q, 
the  determining  condition  being  that  the  deflections  of  the  beam 
at  D  and  C  shall  be  equal.  Tiiis  is  the  effect  of  the  portal 
bracing  when  taken  as  absolutely  rigid  as  compared  with  the 


Fig.  idkb. 

Referring  now  to  Fig.  lo'^b,  we  have  from  the  general  equation  (9), 
page  133,       =  El-^i  for  any  section. 
For  X  <iz. 


Fig.  2o8(z. 
deflections  of  the  columns 
d'y 


M^^,=  EI 


X). 


For  the  point  of  inflection,  where  x  —  x^,  we  have 

M^^^o  =  R{c  -  X,)  -  Q{z  -  X,)  

Integrating  eq.  (13)  between  the  limits  o  and  z,  we  have 

^^%= ^/(^  -  -  Q/y  -  = ^  (-  -  f ) 

which  is  EI  times  the  angle  of  the  deflected  column  at  D. 
For  X  z, 

dy 


<2: 


(13) 


(14) 


(15) 


and  we  have  for  the  portion  beyond  D, 
^dy 


EI 


dx' 


R{c  -  x\ 


^^'J^  =  ^"g^^      D  from  (15)  +  angular  change  beyond  D, 


(16) 


Integrating  this  again  from  z  to  c,  we  obtain  the  deflection  at  C  as  compared  to  that  at  D. 


i6o  MODERN  FRAMED  STRUCTURES. 

But  by  the  conditions  of  the  problem  D  and  C  deflect  equally,  and  hence  this  last  integral 
IS  zero,  or  from  (i6), 


^'y  =  ^/(-f)-  -       =  +  <  -f) =o-.  c.;) 

Q~  2e  ^  2CZ  -      '     °    °  =  o  (I8) 


whence 

R 


But  from  (14)  we  have 

R    ^  —  -y, . 

•    .   .   •   .    o   .   c   c   c   ,   c   .   .  (19) 

whence 

2li7T^/-  (20) 

This  gives  the  point  of  contraflexure  at  which  //and  F(as  in  Art.  115)  are  to  be  applied. 
If  P  is  the  total  applied  external  force  at  C,  as  in  Fig.  208^?,  we  have 

//  =  f,    and     V^P^^^j^.  ..........  (21) 

Maximum  bending  moment  on  column  (at  its  base)  =  Hx^  I 
Bending  moment  at  =  H  {z  —  x^;\'    '  ' 

since  the  horizontal  and  vertical  reactions,  //and  V,  must  be  supposed  to  act  at  the  point  of 
inflection. 

The  remaining  analysis  is  exactly  the  same  as  that  given  in  Art.  1 15,  using  here  c  ~  in 
place  of  c. 

Thus,  with  centre  of  moments  at  D,  we  have 

Stress  in  CC'=/'+^(^°)  =  ^(i  +  ^")  «  (23) 

With  centre  of  moments  at  C,  we  have 

Stress  in /?/)'  =  // ......    .0.  (24) 

From  2  vertical  components  =  o  we  have 

Stress  in  C'Z;  =  F sec  ^=  F—^-.  .    .........  (25) 

e 

If  Z  =  compressive  stress  in  the  column  from  the  vertical  loading,  and  0  =  angle  column 
makes  with  the  vertical,  then 

L  -\-  V  sec  (f>  =  total  direct  stress  in  A'C, 

and 

L  —  Fsec  (p  =  total  direct  stress  in  AC. 

.  p 

The  maximum  bending  stress  in  the  column  by  (22)  is  at  the  base  and  is  — x^. 
From  eq.  (12)  we  then  have  for  the  stress  in  the  extreme  fibre  /ro7n  bending 

P 

f  _         ^  ?   (26 ) 

^      6E  6E 
The  requisite  strength  of  anchorage  may  be  computed  from  the  bending  moment  at  the 
base  of  the  column  (eq.  (22)). 

This  problem  is  usually  solved  by  assuming  the  point  of  inflection  ^z  from  the  base.  For 

*  Since  xo  is  always  greater  than  ^,  it  follows  that  the  bending  moment  at  the  base  of  the  column  is  always  greater 
than  at  D. 

+  Here  the  coefficient  is  -  instead  of  — ,  as  in  eq.  (12),  since  the  deflection  is  that  of  a  beam  of  length  -  fixed  at  one 
'  6  10  2 

end  and  loaded  at  the  other. 


COMBINED  DIRECT  AND  BENDING  STRESSES. 


i6i 


the  extreme  case  where  e  ~  z  —      eq.  (20)  gives     =         When  e  is  less  than  this  the  point 

of  inflection  approaches  the  middle  point  between  A  and  D,  so  that  it  may  be  said  this  point 

z         %  ... 
lies  somewhere  between  -  and  -  z  above  the  base  for  all  cases  of  fixed  base.    But  this  base  is 

2  8 

never  perfectly  fixed  in  direction,  and  any  flexibility  here  would  lower  the  point  of  inflection. 
Neither  is  the  web  bracing  above  perfectly  rigid,  and  any  distortion  here  would  raise  the  point 
of  inflection,  so  that  these  assumptions  may  be  considered  as  offsetting  each  other,  and  the 
formulae  applied  rigidly  as  above. 

152.  The  Trussed  Beam, — A  wooden  beam  is  often  trussed  as  shown  in  Fig.  209. 
There  are  usually  two  tie-rods  passing  outside  the  beam  and  having  their  nuts  bearing  upon  a 
cast-  or  wrought-iron  end-plate.  At  the  middle  a  strut-piece  or  king-post  is  inserted,  of  any 
desired  length.  The  beam  being  continuous,  the 
load  carried  at  C  will  be  -I  of  the  total  uniformly 
distributed  load,  or  \%vl,  where  iv  =  load  per  linear 
inch  of  beam  and  /  =  the  half-length  AC.  The 
greatest  bending  moment  in  the  beam  occurs  at 
C,  and  hence  here  is  to  be  found  the  greatest  fibre 
stress  from  bending.  For  reasonable  depths  of 
truss,  H,  the  deflection  of  C  would  be  inappreciable 
and  may  be  neglected  in  computing  The  ordinary  solution  will  hold  here,  therefore,  or 
the  fibre  stress  may  be  found  for  the  cross-bending  alone  and  added  to  that  from  the  direct 
compression. 

We  have,  for  the  bending  moment  at  C, 

or  /,  - 


for  a  beam  of  solid  rectangular  section. 

The  direct  compression  in  the  beam  is 

5  ti>r 

Therefore 


/ 


P  = 


or 


J/i 


5  w/' 

sl7F/r 


(27) 


(28) 


Zbh'H 


'6//+ 5/0   o   .   .   .  (29) 


The  stress  in  the  tie-rod 


5  wl 


(30) 


The  Queen-post  Truss  without  Counters. — If  the  counter-struts  shown  by  dotted  lines  in 
Fig.  210  id)  be  omitted,  as  is  often  done  in  unscientific  construction,  especially  when  a  beam 
is  trussed  from  below  by  a  tie-rod  and  two  posts,  any  want  of  symmetry  in  the  loading  pro- 


Fig.  210. 

duces  a  bending  of  the  loaded  beam,  as  shown  in  Fig.  210  {d). 

Let  it  be  assumed  that  the  beam  is  supported  by  the  tie-rods  at  points  -^/from  each  end. 
Place  the  load  W  dX  one  of  these  points  and  find  the  stresses  in  all  the  members. 


l62 


MODERN  FRAMED  STRUCTURES. 


Let  H  ~  height  of  truss  ,  k  =  height  of  bottom  chord  ;  b  =  breadth  of  bottom  chord ; 
length  of  bottom  chord  between  end  joints.    Since  the  combination  acts  as  a  whole  to 
carry  the  load  W  to  the  abutments,  the  same  as  any  beam,  the  supporting  forces  are 

it,  =  —  ;    K.  =  : 

3  3 

It  is  evident  at  once  that  the  load  W  divides  itself  between  the  truss  and  the  bottom 
chord  acting  as  a  beam,  and  that  the  truss  will  distort  as  shown  in  Fig.  210  (d).  A  part  of  W 
therefore,  is  carried  by  the  tie-rod  DF,  and  the  remainder  rests  on  AB  at  as  on  a  beam.  To 
find  the  relative  value  of  these  two  portions  into  which  W  is  divided  we  have — 

1.  Since  the  distortion  is  small  and  CD  remains  sensibly  horizontal,  and  the  two  inclined 
members  A  C  and  DB  are  sensibly  of  equal  inclination,  the  horizontal  components  of  these  latter 
are  equal,  since  they  are  each  equal  to  the  stress  in  CD.  But  being  of  equal  inclination  to 
the  horizon,  their  vertical  components  of  stress  must  be  equal. 

2.  But  the  vertical  components  in  ^Cand  DB  are  equal  to  the  stresses  in  the  tie-rods 
CE  and  DF,  respectively,  and  therefore  the  stresses  in  these  rods  are  equal. 

3.  From  the  symmetry  of  the  trussing,  if  the  point  D  drops  a  certain  amount  the  point 
C  must  lift  by  the  same  amount,  and  hence  the  bottom  chord  is  deflected  upwards  at  E  as 
much  as  it  is  downwards  at  F,  and  therefore  the  forces  producing  these  deflections  are  equal. 
But  the  force  producing  the  upward  deflection  at  E  is  the  stress  in  the  tie-rod  EC,  while  the 
force  producing  downward  deflection  at  F  \?>  the  remaining  portion  of  {Rafter  the  part  taken 
by  the  tie-rod  FD  is  subtracted.  That  is,  the  part  of  coming  Hirectl}-  upon  the  beam  is 
just  equal  to  the  stress  in  the  tie-rod  EC,  and  therefore  equal  to  that  in  FD. 

Therefore  the  load  divides  itself  into  two  equal  parts,  one  half  being  carried  by  the 
truss  and  the  other  half  by  the  bottom  chord  acting  as  a  beam. 

W 

The  stress  in  the  tie-rods  CE  and  DF  is  therefore  — ,  and  this  is  the  vertical  component 
in  each  end-post.  Hence 

Compression  in  CZ> )  _  ^     ^  _ 

Tension  \x\  AB        )  "  T  '  3^  ~  ^  

Compression  in  AC  and  DB  =   (32) 

Since  the  central  point  of  the  bottom  chord  is  a  point  of  inflection  in  the  bent  beam,  there 
is  no  bending  moment  at  this  point,  and  it  may  be  treated  as  a  free  supported  end.*  The 

W 

moment  at  F  is  then,  for  a  load  —  at  F, 

2  ' 

^■-18  ^^  =  w  ^^^^ 

The  direct  stress  in  AB  from  (31)  is 

Wl  Wl 
^■^  =  6miC  f  =  f-\-f  =  -^^jf2H^h\  (34) 

which  gives  the  maximum  tensile  stress  per  square  inch  in  the  bottom  chord. 

*Or  a  vertical  section  may  be  passed  through  this  point  of  inflection  and  that  point  taken  a  centre  of  moments, 

2  /  / 
-IV -—  IV- 

3  2  6  Wl 

whence  the  stress  in  CD  =  =  — -  =  also  tension  in  AB.    Also  the  shear  at  the  point  of  inflection  in  AB 

H  (yli 

1         W  W   I  Wl 

—  w  W  —  — ,  whence  the  moment  at  F  and  E  =  —  .  -  =  .    This  method  is  applicable  to  any  truss  symmet- 

3  3  3     6  18 

rical  about  the  panel  where  the  bracing  is  omitted. 


COMBINED  DIRECT  AND  BENDING  STRESSES. 


SECONDARY  STRESSES. 

153.  Definition. — Secondary  stresses  are  those  bending  stresses  arising  from  such  causes 
as  the  following : 

1.  The  members  coming  together  at  a  joint  do  not  have  their  centre-of-gravity  lines 
meeting  in  a  point. 

2.  Members  are  not  loaded,  or  attached,  symmetrically  with  reference  to  the  centre-of- 
gravity  lines. 

3.  Members  are  not  free  to  rotate  at  the  joints  when  the  live  load  comes  on,  and  hence 
they  must  spring  or  deflect  as  the  structure  deflects. 

A  few  of  the  more  common  cases  will  be  investigated. 

154.  Gravity  Lines  not  Meeting  at  a  Point. — One  of  the  more  common  instances  of 
this  fault  can  be  found  in  shallow  lattice  girders,  like  the  New  York  Elevated  Railroads.  Here 
the  single  intersection  Warren  girder  riveted  trusses  have  joints  as  shown  in  Fig.  211.  The 
gravity  lines  of  the  web  members  intersect  in  C,  while  the 
intersections  with  the  upper  chord  are  in  A  and  B.  This 
gives  rise  to  a  bending  moment  at  this  joint  equal  to  the 
pull  in  BD  into  the  arm  A  C,  moments  being  taken  about  A. 
Let  the  chord  be  made  up  of  two  3  in.  X  4  in.  angles,  25 
lbs.  per  yard,  and  one  plate  12  in.  X  ^in.  Let  the  diagonals 
be  composed  of  two  3  in.  X  4  in.  25  lb.  angles  each,  all 
riveting  coming  upon  the  4-in.  leg.  If  we  assume  a  unit 
stress  of  7500  lbs.  in  these  web  members,  or  a  total  stress 
of  37,500  lbs.,  and  if  the  inner  (4  in.)  legs  of  the  angles 

meet  on  the  web  of  the  chord  as  shown  in  the  sketch,  then  the  leverage  AC  would  be  in. 
The  bending  moment  at  this  joint  would  then  be  5^^  X  37,500  =  206,000  inch-pounds. 

If  the  truss  be  assumed  5  feet  high  and  the  web  members  at  45°,  then  these  members 
are  7  feet  long  and  the  panel  lengths  are  5  feet  long.  Adjacent  joints  in  one  chord  are  sub- 
jected to  moments  of  like  sign,  and  all  members,  both  web  and  chord,  are  bent  in  opposite 
directions  at  their  opposite  ends,  and  by  the  same  amounts  approximately.  This  puts  them 
all  in  double  curvature,  and  makes  a  point  of  contrary  flexure  occur  at  their  centres.  All 
members  meeting  at  a  joint,  therefore,  resist  this  bending  action  of  the  members  from  eccen- 
tric position,  in  proportion  to  their  relative  rigidities.  The  angular  movement  of  the  joint 
would  be  the  same  for  all  the  members  meeting  there  and  would  be  resisted  by  all,  acting  as 
beams  fixed  at  one  end  (the  joint)  and  free  at  the  other  (the  middle  section  of  the  member 
where  the  moment  is  zero,  it  being  a  point  of  contrary  flexure).  The  angular  change  at  the 
joint  would  then  be  the  deflection  of  the  points  of  inflection  (middle  points),  divided  by  the 
half  lengths  of  the  members.     Whence  if  /  =  this  half  length,  we  have  for  the  angle 

pr         PI'  Ml 

through  which  the  joint  has  turned,  a  =  /  =         =         where       is  the  moment  at 

the  joint  carried  by  one  member.    Whence  we  find 

M.  =  '—r  (35), 

Since  E  and  a  are  the  same  for  all  the  members,  we  see  that  the  several  members  meet- 
ing at  a  rigid  joint  will  share  the  total  bending  moment  developed  at  that  joint  directly  as 

their  moments  of  inertia  and  inversely  as  their  lengths,  or  as-j .  To  divide  this  moment  prop- 
erly among  the  members  meeting  at  the  joint  and  rigidly  attached,  we  have  only  to  take  out 
J  for  each  of  these  members  and  divide  the  total  moment  amongst  them  in  proportion  to 
these  values. 


164 


MODERN  FRAMED  STRUCTURES. 


In  the  example  taken  we  have  for  the  moment  of  inertia  of  the  chord  137.6,  the  length 
of  a  chord  member  being  5  feet.  The  moment  of  inertia  of  one  web  member  (two  angles),  in 
the  plane  of  the  longer  leg,  is  8.0  and  its  length  is  7  feet. 

We  have  then 

/ 

For  each  chord  member  j  =  27.5  ; 

/ 

"      "    web        "      J  =  I.I. 

i' 

There  being  two  members  of  each  kind  meeting  at  this  joint,  the  total  sum  is  57.2,  Therefore 
each  chord  section  takes  |  =  48  per  cent,  and  each  web  member  takes  -^^j  =  2  per  cent  of 
the  moment  coming  at  the  joint. 

This  result  shows  at  once  that  we  might  have  assumed  at  the  start  that  the  moment 
was  wholly  resisted  by  the  chord  section  without  appreciable  error. 

The  total  moment  at  the  joint  has  been  found  to  be  206,000  inch-pounds.  Forty-eight 
per  cent  of  this  is  98,880  inch-pounds.  This  is  the  bending  moment  resisted  by  the  chord 
section.  The  stress  on  the  extreme  lower  fibre  of  the  web  of  this  member,  which  lies  8.16 
inches  from  the  neutral  axis,  is 

^      M,:y,     98,880  X  8.16       ^  ,^ 
7,  =  — =  137  6         ~  square  mch 

in  the  extreme  bottom  fibres  of  the  top  chord,  being  tension  on  one  side  of  the  joint  and 
compression  on  the  other. 

Since  the  chord  stress  for  which  this  member  was  designed  was  probably  about  7500  lbs. 
per  square  inch,  we  see  that  the  secondary  stress  with  this  apparently  favorable  arrangement 
of  the  web  connection  gives  rise  to  secondary  stresses  yS  per  cent  as  large  as  the  primary 
stresses  which  were  alone  taken  into  account  in  the  computation.  In  other  words,  the  actual 
maximum  fibre  stress  is  nearly  twice  as  great  as  that  for  which  the  structure  was  designed. 
Since  it  is  not  uncommon  to  find  riveted  structures  with  joints  much  more  eccentric  than  the 
one  here  computed,  the  importance  of  avoiding  such  eccentric  combinations  is  patent. 

In  the  computation  of  secondary  stresses  arising  from  the  non-concurrence  of  the  gravity 
lines  of  the  assembled  members  no  great  error  will  be  made  if  it  be  assumed  that  the  heavier 
and  more  rigid  members  take  all  the  moment. 

In  computing  this  moment  take  a  centre  of  moments  at  the  intersection  of  all  the  forces 
but  one  (if  possible),  and  then  the  moment  developed  is  the  remaining  force  into  its  arm.  If 
a  centre  of  moments  cannot  be  chosen  so  as  to  leave  out  but  one  force,  then  the  products  of 
all  the  remaining  forces  into  their  several  arms  are  found  and  the  algebraic  sum  taken.  This 

moment  is  then  divided  amongst  the  members  in  proportion  to  the  quantity  j  for  the  several 

members.  If  some  of  the  members  are  free  to  turn  at  the  other  ends,  then  their  full  lengths 
are  to  be  taken,  while  for  those  members  which  are  rigid  at  both  ends  one  half  their  lengths 
are  used  for  /.  If  all  members  are  alike  in  this  particular,  either  all  free  or  all  rigid  at  their 
opposite  ends,  then  their  full  lengths  can  be  used,  as  the  reactions  will  be  the  same  whether  the 
full  lengths  or  the  half  lengths  be  taken.  The  moments  of  inertia  must  be  computed  for  the 
sections  at  or  near  the  joints. 

155.  Members  not  Loaded  Symmetrically  with  reference  to  their  Centre  of 
Gravity  Lines. — This  is  a  simple  case  and  one  not  requiring  any  extended  discussion.  The 
bending  moment  is  in  all  cases  equal  to  the  total  load  (pull  or  thrust)  on  the  member  into  its 
arm,  which  is  the  perpendicular  distance  between  the  centres  of  gravity  of  the  applied  load  and 
of  the  cross-section  of  the  member.  The  bending  will  be  in  the  plane  of  this  lever  arm,  and  the 
stress  resulting  from  this  eccentric  loading  is  given  by  the  general  equation  (4),  the  negative 
sign  in  the  denominator  to  be  used  when  the  member  is  in  compression  and  the  positive  sign 
when  in  tension. 


COMBINED  DIRECT  AND  BENDING  STRESSES. 


156.  Example  8.  What  is  the  greatest  secondary  stress  in  one  of  the  angle-irons  shoivn 
in  Fig.  211,  due  to  its  being  attached  by  one  leg  only,  when  the  compressive  load  upon  it  is  7500 
lbs.  per  square  inch  or  18,750  lbs.  on  one  angle. 

Equation  (4)  now  takes  the  form 

^  M,y, 

Since  the  load  P  may  be  supposed  to  be  concentrated  at  the  centre  line  of  the  wide  leg, 
which  is  riveted  to  the  chord,  the  eccentricity  is  therefore  for  this  angle  iron  but  0.61  inch. 
The  moment  is  0.61  X  18,750  =  11,440  inch  pounds  =  yl/, .  The  distance  from  the  centre-of- 
gravity  axis  of  the  angle  to  the  most  compressed  fibre  is  0.8  inch,  and  /  =  1.94.  Hence  we  have 

^              11,440  X  0.8                9150         ^  •  u 

/,  =  —  -r-  =  -—         —  0220  lbs.  per  square  mch. 

18.750  X  7056     1.94  — .47 

^■^"^  280,000,000 

If  the  two  angles  forming  one  member  are  attached  together  at  intervals  throughout  their 
lengths,  they  would  mutually  support  each  other,  since  they  would  tend  to  deflect  in  opposite 
directions.  This  would  prevent  the  development  of  the  secondary  stresses  here  computed. 
For  this  reason  angle-irons  when  attached  by  one  leg  only  should  always  be  placed  in  pairs, 
and  then  attached  together  throughout  their  entire  lengths. 

157.  Example  9.  Find  the  shearing  stress  on  each  rivet  in  the  attachment  shown  in  Fig. 
212.  Let  the  force  /'acting  in  a  direction  parallel  to  rivets  2  and  4  be  15,000  lbs.  This  is  a 
common  case  in  practice  where  a  i-inch  square  lateral  rod  is  held  with 
four  ^-inch  rivets,  giving  3750  lbs.  to  each.  Let  the  eccentricity,  a 
in  the  figure,  be  4  inches.  Then  the  force  P  may  be  replaced  by 
the  equal  force  P'  acting  at  the  centre  of  gravity  of  the  rivets,  and 
a  moment  of  4  X  15,000  =  60,000  inch-pounds.  This  moment  is 
resisted  equally  by  the  four  rivets,  giving  rise  to  equal  shearing 
stresses  in  the  direction  of  the  several  arrows,  or  circumferentially  Fig.  212. 

about  the  centre  of  gravity  of  the  rivets.  The  lever  arm  of  these  forces,  taking  the  rivets  as 
placed  at  the  corners  of  a  square  5  inches  on  a  side,  would  be  3.5  inches.    The  shearing  stress 

on  each  rivet,  therefore,  to  resist  the  moment,  would  be  ^''^^  —  4?oo  lbs.  Therefore 

4  X  3i 

The  shearing  stress  on  rivet  i  =  375°  +  4300  =      8050  pounds. 

"  2  and  4  =  i^(375o)'  +  (43007  =  57 10 
"     3  =3750  —  4300  =—  550 

The  minus  sign  given  to  the  shearing  stress  in  rivet  3  indicates  that  the  shear  on  this  rivet 
is  opposite  in  direction  to  that  in  rivet  i.  The  shearing  stress  on  rivet  i  is  thus  shown  to  be 
more  than  twice  as  much  as  it  was  designed  to  carry,  and  it  is  liable  to  work  loose. 

158.  Secondary  Stresses  due  to  Rigidity  of  Joints.  — Whenever  a  framed  structure 
deflects  under  a  load  all  the  angles  of  intersection  at  all  the  joints  would  be  changed  by  very 
small  amounts  if  the  members  were  free  to  turn  at  the  joints.  To  be  perfectly  free  to  turn 
these  members  would  have  to  be  hung  on  knife-edges.  Pin-connected  bridges  have  generally 
been  supposed  free  to  turn  at  the  joints,  but  this  is  not  the  case.  There  may  be  a  slight 
rocking  of  the  member  about  the  pin  from  the  looseness  of  the  pin  in  the  hole,  but  the 
maximum  play  allowed  now  in  good  work  is  so  small  as  to  be  practically  zero.  For  pin-con- 
nected structures,  however,  the  members  are  comparatively  narrow,  so  that  jj',  in  formula  (4) 
is  small.  For  the  ordinary  proportions  also  of  height  and  panel  length  to  span  the  angular 
change  is  so  small  that  this  source  of  secondary  stresses  may  be  neglected.  In  riveted  work, 
where  wide  plates  are  used,  they  may  be  much  larger.  Such  stresses  need  never  be  computed 
for  pin-connected  bridges. 


i66 


MODE/? AT  FRAMED  STRUCT  URES. 


CHAPTER  XI. 

SUSPENSION-BRIDGES. 

159.  Introduction. — Suspension-bridges  of  a  crude  form  have  been  in  use  from  the 
earhest  times.  The  first  long-span  iron  bridges  were  also  of  this  type,  the  suspension-cables 
being  composed  of  chains,  iron  bar  links,  flat  bars  with  pin  connections,  and  finally  of  iron 
and  steel  wire.  Sir  Thomas  Telford  and  Sir  Samuel  Brown  greatly  developed  and  improved 
the  designs  of  suspension-bridges  in  England  from  1814  to  1830,  while  Mr.  J.  A,  Roebling 
adapted  this  style  of  structure  to  both  highway  and  railway  service  in  his  famous  bridge  over 
the  Niagara  River  in  1852-5.  Telford's  and  Brown's  bridges  were  built  without  the  use  of 
stays,  as  shown  in  Fig.  213.    This  is  a  span  of  432  feet,  built  by  Mr.  Brown  in  1829,  the  cable 


Fig.  213. 

being  composed  of  flat  wrought-iron  bars.  Fig.  214  is  a  view  of  one  half  of  the  Niagara  bridge 
as  it  was  originally  constructed,  the  span  being  821  feet.    Diagonal  stay-cables  are  introduced. 


Fig.  214. 


and  the  trusses  are  exceptionally  strong,  having  a  height  of  18  feet  out  to  out.  The  sag  of 
one  cable  is  54  feet,  and  of  the  other  64  feet  ;  each  is  composed  of  3640  No.  9  iron  wires.* 

When  the  suspension-cables  are  made  of  a  high-grade  steel  wire  having  an  average  ulti- 
mate strength  of  some  160,000  lbs.  per  square  inch,  and  which  may  be  dimensioned  for  a  maxi- 
mum working  load  of  40,000  lbs.  per  square  inch,  there  is  great  economy  in  this  style  of  con- 
struction. When  the  material  is  disposed  in  small  wires,  however,  presenting  a  very  large  ratio 
of  surface  to  sectional  area,  it  must  be  protected  more  perfectly  from  corrosion  than  is 
necessary  when  the  material  is  disposed  in  larger  members.  When  so  protected,  a  long  span 
can  be  built  on  this  principle,  for  highway  trafific,  for  a  much  less  sum  than  would  be  required 
for  any  form  of  truss-bridge  to  carry  the  same  loads.  To  stiffen  a  suspension-bridge  suf- 
ficiently to  carry  train  loads  adds  so  greatly  to  the  cost  as  to  make  it  inadvisable  to  employ 

*This  bridge  has  now  been  reconstructed  by  Mr.  L.  L.  Buck,  M.  Am.  Soc.  C.E.,  by  replacing  the  stone  towers  by 
steel  construction,  the  "wooden  stiffening  truss  by  one  of  steel,  and  omitting  the  stays  altogether. 


S  USPENSION-BRID  GES. 


167 


it  for  railway  purposes  except  for  the  very  longest  spans,  as  in  the  case  of  the  proposed 
bridge  over  the  Hudson  River  at  New  York  City.  Here  the  span  is  2850  feet,  and  provision 
is  to  be  made  for  carrying  six  tracks,  with  a  possible  increase  to  ten  tracks  if  required.  For 
highway  purposes  suspension-bridges  can  be  judiciously  used  for  spans  exceeding  300  or  400 
feet.  If  the  locality  is  adapted  to  the  use  of  metallic  arches,  these  could  well  be  employed 
for  spans  of  from  300  to  500  feet.  For  all  spans  of  greater  length  than  500  feet  for  highway 
purposes  the  suspension  type  of  bridge  should  perhaps  always  be  employed. 

THEORY  OF  SUSPENSION-BRIDGES. 

160.  Stress  in  Cable  for  Uniform  Load  over  the  Entire  Span. — For  this  loading  the 
curve  of  equilibrium  is  a  parabola  as  shown  in  Art.  49.    It  may  be  proved  directly  as  follows: 


I 


PL       .        .  .  . 

'.if 

Fig.  215. 


Let  Fig.  215  {d)  represent  a  suspension-bridge  uniformly  loaded  with  a  dead  load  w  and  a 
live  load  p  per  foot.  Let  /  =  span,  and  v  —  sag,  or  versed  sine  of  curve.  In  Fig.  (^)  let  the 
left  half  of  the  bridge  be  removed  and  replaced  by  the  force  H,  this  being  the  stress  in  one 
cable  at  bottom  where  it  is  horizontal.  Let  O  be  taken  as  the  origin  of  co-ordinates.  Take 
moments  about  any  point  in  the  cable  as  P,  whose  ordinates  are  PD  =y,  and  OD  =  x.  Then 
we  have,  for  the  external  forces  acting  on  the  left  of  the  section  through  P,  since  the  moment 
here  is  zero, 


Hy=\ 


2~l'2' 


or 


(I) 


which  is  the  equation  of  a  parabola  referred  to  its  vertex. 

To  find  H  take  moments  about  A,  the  top  of  the  tower,  and  we  obtain 


H.=  ^-±t%  or  H=m^)^, 


(2) 


Using  this  value  of  H"m  eq.  (l)  we  have,  as  the  equation  of  the  curve  of  equilibrium  for 
a  load  uniformly  distributed  along  the  horizontal,  referred  to  its  lowest  point, 


(5} 


i68 


MODERN  FRAMED  STRUCTURES. 


In  such  a  curve  of  equilibrium  the  stress  is  simple  teusion,  and  the  horizontal  component  of 
this  stress  is  constant.  This  must  be  so  since  the  external  forces  acting  on  the  cable  are  all 
vertical,  having  no  horizontal  components.  Hence 


The  tension  in  cable 


=  H  sec  z  =  1/  7^  =  //  ^ 

ED  ax 


The  tension  in  cable  at  totvers  =  H  sec  i  — 


IV  -|-  /' 


(4) 
(5) 


where  i  =  angle  of  cable  with  the  horizontal. 

i6i.  StifTening  Truss  for  Partial  Loads. — When  the  cable  is  not  loaded  uniformly 
over  its  entire  span  it  will  swing  into  a  new  curve  of  equilibrium  for  that  load,  if  this  action 
is  not  resisted  in  some  way.  It  is  resisted  by  a  truss  either  along  the  roadway  or  attached 
to  the  cable  itself,  or  both.    The  object  of  the  truss  is  only  to  distribute  the  load  uniformly 


Fig.  2i6. 

•over  the  cable,  and  not  to  assist  in  carrying  it  to  the  abutments.  The  truss  should  be  made  to 
deflect  so  little  that,  as  compared  with  the  free  movements  of  the  cable,  it  will  be  relatively 
rigid  ;  that  is  to  say,  its  deflection,  up  or  down,  at  any  point  under  a  concentrated  load  will  be 
very  small  as  compared  to  the  deflection  of  the  cable  at  that  point  under  this  loading,  if  the 
truss  were  not  used.  Whenever  the  truss  is  reasonably  efficient  in  preventing  unequal  dis- 
tortions this  condition  of  relative  local  rigidity  holds  true,  and  hence  the  conclusions  based 
on  this  assumption  are  correspondingly  correct. 

In  the  following  discussion  the  load  is  supposed  to  rest  primarily  upon  the  truss,  as  shown 
in  Fig.  2i6,  and  the  truss  is  supposed  to  distribute  this  load  evenly  or  uniformly  upon  the 
cable.  Whatever  the  contingencies  of  loading,  therefore,  the  stress  in  the  hangers  is  assnmed 
to  be  constant  from  end  to  end  of  the  span. 

A  further  assumption  is  that  the  cable  stretches  very  little  for  any  given  unsymmetrical, 
load,  so  that  when  this  load  is  uniformly  distributed  over  it  by  the  truss,  its  vertical  deflection 
at  the  centre  is  insignificant  as  compared  to  the  vertical  deflection  of  the  truss  acting  alone 
under  this  same  total  load.  The  former  assumption  referred  to  a  deflection  of  the  cable  at  a 
given  point,  under  a  concentrated  load  there,  due  to  deformation,  or  change  of  shape,  if  acting 
alone,  while  this  assumption  has  to  do  only  with  the  symmetrical  deflection  of  the  cable  due 
to  its  elongation.  The  deflection  of  the  truss  is  very  small  as  compared  to  the  former,  and 
very  large  as  compared  to  the  latter. 

If,  therefore,  the  cable  does  not  appreciably  increase  its. sag,  for  the  assumed  distribution 
of  the  concentrated  loads,  then  the  cable  may  be  assumed  to  carry  all  this  live  load,  while  the 
truss  merely  serves  to  distribute  it.    But  the  condition  of  equilibrium  of  the  external  forces 

'i  i 


Rl 


Fig.  217. 

acting  upon  the  truss  requires  that  the  sum  of  their  vertical  components  and  of  their  moments 
shall  be  zero. 


SUSPENSION-BRIDGES.  169 
The  forces  acting  on  each  truss  are : 

P 

1st.  A  uniform  downward  load  of  —  lbs.  per  foot  over  the  distance  x  from  the  right 
support. 

2d.  A  uniform  upward  pull  of  q  lbs.  per  foot  from  each  cable,  over  the  entire  span. 
3d.  The  two  end  reactions,  A',  and      ,  for  equilibrium.    We  have  assumed  that 

—  =  ql,    or    q=-^  (6) 

This  satisfies  the  condition  of  equality  of  vertical  components  of  external  forces  without 

px  X 

reference  to  the  end  reactions.    But  the  centre  of  gravity  of  the  downward  forces  —  is  —  from 

B,  Fig.  2 1 7,  while  that  of  the  pull  of  the  hangers  is  at  the  centre  of  the  span.    These  being  equal 

I  —  X 

and  opposite  non-concurrent  forces,  they  form  a  couple  whose  arm  is  — ^ — ,  the  moment  being 
pxil —  x\ 

—  ^ — ^ — 1.  This  couple  can  only  be  balanced  by  another  couple  composed  of  the  end 
reactions  having  the  arm  / ;  hence  these  reactions  are  equal  and  opposite,  and  equal  to 

or 

=  + 7?,   (7) 

We  now  have  all  the  external  forces  which  act  upon  the  truss  and  are  prepared  to  find 
the  maximum  moments  and  shears  coming  upon  it,  for  the  uniformly  distributed  moving  load 
p  per  running  foot.  Since  the  end  reactions  are  both  positive  and  negative  for  different 
loadings,  the  truss  must  rest  upon  the  piers  and  be  anchored  to  them  sufificiently  to  resist  the 
maximum  negative  end  shear  or  reaction.  We  will  first  find  what  jjosition  of  load  gives 
maximum  shears  and  moments,  and  then  find  values  for  these  maxima. 

Let  the  load  extend  from  the  right  support  to  a  distance  x  towards  the  left,  as  shown  in 
Figs.  216  and  217. 

Let  s  be  the  distance  from  the  right  support  to  any  section,  so  that  ,c  may  be  greater  or 
less  than  x,  and  may  vary  from  o  to  /. 

162.  Discussion  for  Maximum  Shear. — The  general  equations  for  shear  (positive  shear 
being  that  which  acts  upwards  on  the  left)  are,  for  one  truss, 


S,^^=-R^^q{l- z)-^^{x-z\ 


5;>  ^  =  -    +  q{i  -  4 

px  i)X  II   x\ 

Substituting  ^  for  q  and  ^— ^ — j  for  7?, ,  we  have 


(8) 


S.<.=  ^{x-l-2a)+^;  (9) 

S,>.=  ^{x  +  l-22)  (10) 


I70  MODERN  FRAMED  STRUCTURES. 

px 

If  we  take  first  a  given  load,        and  let  it  remain  stationary  while  our  section,  z,  varies 

from  o  to  /,  using  eq.  (9)  from  o  to  x,  and  eq.  (10)  from  x  to  /,  we  see  by  inspection  that  the 
shear  is  a  minimum  (neg.  max.)  iox  z  =  o  and  for  z  =  I,  and  a  positive  maximum  for  z  =  x. 
Thus  for  ^  =  o  and  z  =  I -wq  have,  from  eq.  (9)  and  (10), 

px 

^z  =  o—  ^J^i-^  —  ^)  —  Sz  =  I       •     •     •     •     •     •     •     •      .     •  (11) 

for  negative  shear, 

px 

and  iox  z—  x  we  have  5^=^  =  ^(/  —  x")  (12) 

for  positive  shear.  From  these  equations  we  see  that  the  shear  at  the  head  of  the  load  is 
ahvays  nutnerically  equal  to,  but  of  opposite  sign  from,  the  shears  at  the  ends  of  the  span,  and 
that  these  are  the  maximum  shears  on  the  truss  for  a  continuous  load  from  one  support. 

If  we  let  the  load  move  across  the  span,  making  x  vary  from  o  to  /,  we  can  find  from 
eqs.  (11)  and  (12)  the  positions  of  load  giving  maximum  positive  and  negative  shear. 

Thus,  from  either  eq.  (i  i)  or  eq.  (12),  we  have 

dS  I 

-J-  =  o  =  2x  —  I,  or    X  =  —  for  shear  a  maximum, 
ax  2 

or  the  greatest  shears  all  occur  when  the  bridge  is  half  loaded. 

Substituting  this  value  of  x  in  either  of  the  eqs.  (11)  or  (12),  we  find  the  Maximum  Posi- 
tive and  Negative  Shears  in  one  truss  at  both  the  ends  and  the  centre  to  be  -^^pl ;  or,  in  general. 

Maximum  Shears  =  -itpl-  (13) 

To  obtain  the  maximum  positive  shear  at  any  point  distant  z  from  the  right  support,  we 
must  study  the  three  conditions  indicated  by  equations  (9),  (10),  and  (12).  Thus  for  the 
maximum  shear  at  the  head  of  the  load,  putting  z  =  kl  to  adapt  our  results  to  all  spans, 
we  have 

S.=.  =  ^^{4^-4^^)   (A) 

pi 

=        for  graphical  representation  in  Fig.  217a:*. 


This  locus  is  the  parahola.  A  CB,  while  for  a  load  from  the  left  the  shear  would  be  negative 
and  given  by  the  curve  AC'B. 

To  find  the  maximum  negative  shear  for  the  right  half  of  the  span,  when  the  load  comes 
on  from  the  right,  we  have  z  <Zx  as  given  in  eq.  (9).  To  find  the  position  of  this  load  which 
will  give  a  maximum  shear  of  this  kind,  at  any  point  z  under  the  load,  we  must  solve  (9)  for 
6"  a  maximum  for  x  variable,  and  obtain 

dS^<:,  l-\-2z 

— —  =.  o  z=  2x  —  /  —  2z,     or     X  =  . 

dx  2 


*  The  diagrams  in  this  figure  arc  due  to  Prof.  M.  A.  Howe. 


S  USPENSION-BRID  GES. 


170a 


Substituting  this  value  of  x  in  (9)  and  putting  z  =  kl,  we  find 

pi 

Max.  5^<^  —  --^{Ak  —  4^'  —  i) 


(B) 


=^(F  —  i)  for  graphical  representation. 


This  is  the  curve  EF'  in  the  figure.  Evidently  for  a  load  from  the  left  we  would  have 
had  similarly  the  positive  shear  curve  DE. 


1.5 

Q 

K  " 

1.5 

1.2 

H 

1J2 

Loa 

\l-—f 

\ 

c 

m 

a** 

0.6 

M  =  j 

0.6 

S=i 

s 

> 

A 

0.3  / 

— 

1  / 
k' 

0 

8 

'■""■-e 

6--. 

E 

.— e 

0 

2 

lilies 

"0/ 

fc.  = 

_  2 
"  i 

« 

0.3\ 

^-^ 

M 

M 

0.6 

MONf 

ENT 

AND 

 1 

SHE/ 

-OiLA 

1  

iR  Dl 

AQR/ 

MS 

— ^ 

-OJi 

SI 

SPEr 

isior 

BRI 

3QE 

FRUS 

S 

t£>>. 

0.9  ^ 

A' 

c' 

ft 

-  =a; 

— Lot 

 . 

fXQJ>l 

1.8 

— hH- 

1.5 

M. 


Fig.  217a. 

To  find  the  maximum  negative  shear  on  the  left  half  of  the  span  we  would  have  to  take 
5  >  ;r,  or  take  our  section  in  front  of  the  load  coming  on  from  the  right.  This  is  given  in 
equation  (10).    Treating  this  as  before,  we  find 

dS:,>^  2Z  —  1 

— J —  =  o  =  2x  -\- 1  —  2z,     or     X  =   , 

dx  '  2 


which  substituted  in  (10),  and  putting  z  =  kl,  gives 


Max.  5^>^  =  76^4^     4-^'  —  0. 


(C) 


which  is  the  same  as  (B).  Therefore  this  locus  U E  is  symmetrical  with  F'E,  and  for  a  load 
coming  on  from  the  left  we  have  FE  symmetrical  with  DE. 


MODERN  FRAMED  STRUCTURES. 


163.  Discussion  for  Maximum  Moments. — Referring  again  to  Fig.  217,  we  may  write, 
for  the  moments  in  one  truss  at  any  section  z, 


and 


i/.>.  =  -^.('--)  +  ^^^^  (15) 


Substituting  the  values  of      and  q,  we  obtain 

^z<.^'j[i   (16) 

M,>.  =  ^{^-  s){x-z)  (17) 

By  inspection  we  see  that  when  ^  =:  or  when  the  section  is  taken  at  the  head  of  the 
load,  the  moment  is  zero  by  both  equations.  Also,  that  when  s  <C  x  the  moment  is  positive, 
and  when  z  >  x  the  moment  is  negative.  Therefore,  i/ie  head  of  tlie  load  is  always  a  point  of 
contrary  flexure.  And  since  we  have  already  learned  that  the  shear  here  is  always  equal  and 
opposite  to  that  at  the  two  ends,  we  see  that  both  the  loaded  and  the  unloaded  portion  of  the 
span  can  be  treated  as  a  simple  truss  uniformly  loaded.  This  leads  us  to  conclude  that  the 
maximum  moments  occur  at  the  middle  parts  of  the  loaded  and  of  the  unloaded  portions,  the 
former  being  positive  and  the  latter  negative  moments.  This  same  conclusion  could  have 
been  reached  by  a  discussion  of  eqs.  (lO)  and  (17),  as  was  done  in  the  case  of  shear  with  eqs, 
(9)  and  (10). 

These  maximum  moments  are 

M     /  +      =   (19) 


2  =  ■ 


To  find  for  what  position  of  load  the  downward  bending  moment  under  the  load  is  a 
maximum,  put  the  first  differential  coefihcient  of  eq.  (18)  equal  to  zero,  and  find 

dM  px     ipx""       ,  ,  ^ 

Doing  the  same  for  eq.  (19)  to  find  position  of  load  for  maximum  upward  bending 
ahead  of  the  load,  we  have 

=  o  z=  r  —  dfXl  +  ix",    whence    x  =  y.  (21) 

Therefore,  the  maximum  dozvmvard  moment  occurs  at  the  middle  of  the  load  zvhen  it  extends 
over  two-thirds  of  the  span,  and  the  maximum  upward  moment  occurs  at  the  middle  of  the  tm- 
loaded portion  when  the  load  extends  over  one-third  the  span. 

Inserting  these  values  of  x  in  eqs.  (18)  and  (19),  we  find 

pl'  ,  . 

Maximum  downward  moment  in  one  truss  =  — 5-;  (22) 

I  OS 

upzvard         "         "        "   (23) 

For  a  load  /  per  foot,  on  two  simple  trusses  of  length  /,  the  bending  moment  at  the  centre 

pr 

of  each  would  be 

16 


S  USPENSION-BRID  GES. 


171 


In  other  words,  the  maximum,  upward  and  downward  moments  are  equal,  occur  at  t'he  one- 
third  and  two-thirds  points,  and  are  about  one-sevc7ith  the  maximum  moment  due  to  the  same  unit 
load  acting  over  the  same  span  when  unsupported  by  the  cable. 

The  maximum  shears  were  found  to  be  -^-^pl,  or  just  one-fourth  what  they  would  be  on 
simple  trusses  of  the  same  length. 

To  obtain  the  maximum  positive  (downward)  moment  at  any  section  under  the  load 
distant  z  from  the  right  support,  find  from  eq.  (16) 

dM,^^  z      2x  ut  \  \ 

—^  =  o=iJ^-  —  ~,     or     x  =  i{l-\-z). 

Substituting  this  in  (16),  we  have 

Max.  M,<^  =  j^^il-  zy  =  ^~k{ I  -  ky     (where  z  =  kl) 

pi'  .  .  .        ■  .  . 

—  for  graphical  representation  in  Fig.  217a.    This  locus  is 

the  curve  AGH  for  a  load  coming  from  the  left  and  BKH  for  a  load  from  the  right. 

For  the  maximum  negative  (upward)  bending  moment  on  the  unloaded  portion  we  have 
from  eq.  (17) 

dM^>^  ,  ,    ,  z 

 -,       —  o  =  2lx  —  2XZ  —  /  +  z*,    or    X  =  —. 

dx  ^    '  2 

Substituting  this  in  (17),  we  have 

Max.  M,^,=  -  -^(/-  z)^-  ^k\i  -  k) 
pP 

=  ^^^F  ior  graphical  representation  in  Fig.  217^:.    This  locus 

is  the  curve  AG' H'  for  a  load  from  the  right  and  BK'H'  for  a  load  from  the  left. 

Since  all  parts  of  the  stiffening  truss  in  a  suspension-bridge  are  subjected  to  both  positive 
and  negative  moments  and  shears,  in  all  parts  of  its  length,  the  maximum  shear  occurring  at 
both  ends  and  at  the  centre,  and  the  maximum  moments  at  the  one-third  points,  it  is  common 
to  dimension  all  parts  to  carry  these  stresses,  thus  making  uniform  sizes  throughout  the  entire 
truss. 

It  should  also  be  noted  that  in  this  case  the  shear  in  the  truss  from  live  load  is  not 
affected  by  the  amount  of  the  dead  load,  this  latter  being  suspended  wholly  from  the  cable. 

If  the  span  is  very  long,  the  actual  maximum  moments  and  shears  may  be  taken  out  for 
all  sections  by  the  use  of  eqs.  (11),  (12),  (18),  and  (19),  and  the  members  proportioned 
accordingly. 

If  the  stiffening  trusses  be  proportioned  to  carry  safely  the  moments  and  shears  here  found, 
there  will  be  no  necessity  for  the  use  of  stay-cables.  In  most  cases  in  practice,  however,  the 
trusses  are  not  as  strong  as  here  assumed  and  the  use  of  stay-cables  becomes  necessary. 

164.  The  Action  of  Stay-cables. — It  is  common  to  use  stay-cables  reaching  from  the 
tops  of  the  towers  to  the  bottom  of  the  stiffening  truss  out  to  about  one-fourth  the  span  from 
the  towers.  At  this  point  the  stays  become  tangent  to  the  main  cables  at  the  towers.  The 
stays  are  superfluous  members,  and  when  introduced  the  distribution  of  the  load  must  be  deter- 


172 


MODERN  FRAMED  STRUCTURES. 


■mined  by  the  principle  that  it  divides  itself  amongst  the  systems  in  direct  proportion  to  their 
relative  rigidity  or  inversely  as  their  deflections. 

We  will  now  find  the  deflection  of  any  point  of  attachment  of  a  stay,  as  B,  Fig.  218, 
resulting  from  assumed  changes  in  unit  stress  in  the  several  members  of  the  systems.  A  load 
placed  at  B  may  pass  up  the  hanger  to  the  cable  and  thence  to  the  tower,  or  it  may  pass 


Fig.  218. 


up  the  stay  to  the  tower,  causing  tensile  stress  in  both  the  stay  and  the  bottom  chord 
of  the  truss  to  the  right  of  B.  Since  the  greatest  proportion  of  the  load  at  B  will  come  upon 
the  stay  when  the  cable  deflects  the  most  at  this  point,  which  will  be  when  the  bridge  is 
fully  loaded,  we  will  assume  this  condition  in  the  following  discussion  : 

The  problem  is,  therefore,  to  find  the  deflection  of  B  in  terms  of  the  increase  in  the  unit 
stress  in  hanger  and  cable  {df^  and  df^  for  a  full  live  load  over  the  whole  span,  and  then  the 
deflection  of  B  in  terms  of  the  increase  in  unit  stress  in  stay  and  bottom  chord  of  truss 
{df^  and  df^  for  the  same  full  load.  Then,  by  placing  these  equal  to  each  other,  since  the  two 
systems  are  rigidly  attached,  decide  how  the  load  at  B  divides  itself  between  the  two  systems. 
It  will  remain  then  to  compute  a  table  of  numerical  coefficients  for  difTerent  positions  of  B 
on  the  outer  quarters  of  the  span.    The  following  notation  will  be  employed : 


Let  / 

length  of  span ; 

V 

sag  at  centre  =  the  vertical  projection  of  the  stays ; 

(xy) 

coordinates  of  the  parabolic  curve  referred  to  0\ 

I. 

length  of  cable  between  towers ; 

Is 

length  of  stay  ; 

4 

length  of  hanger  ; 

/. 

length  of  bottom  chord  of  truss  between  symmetrical  hangers ; 

dy 

deflection  of  cable  at  B  ; 

dv 

deflection  of  cable  at  centre ; 

dh 

stretch  of  hanger  ; 

dv. 

vertical  deflection  at  B  from  stretch  of  stay  ; 

dvt 

vertical  deflection  at  B  from  stretch  of  lower  chord  of  truss  ; 

df^i  df,,,  df, ,  dft 

changes  of  unit  stress  in  cable,  hanger,  stay,  and  truss-chord. 

respectively,  due  to  live  load  ; 

E 

the  common  modulus  of  elasticity  of  cable,  hanger,  and  stay; 

modulus  of  elasticity  of  truss  chord. 

165.  Deflection  of  the  Cable  System  for  Full  Loads. — The  length  of  the  cable,  in 
terms  of  /  and  v,  expressed  by  a  series  and  the  first  two  terms  used,  as  may  be  done  when  v  is 
small  &s  compared  to  /,  is 

i.  =  i^Y-f  :  (^4) 

The  equation  of  the  curve  referred  to  o,  as  an  origin,  is 


S  USPENSION-BRIDGES. 


173 


To  find  the  relation  between  small  changes  of  length  of  cable  and  of  sag  at  the  centre  we 
may  use  the  methods  of  the  calculus.    Differentiating  eq.  (24)  for    and  v  variable,  we  have 

16  V 

dl,=-  —  •  -jdv  (26) 

Doing  the  same  with  (25)  for  j  and  v  variable,  we  have 

dy  =  y^lx  -  x')dv,  (27) 

whence 

dy=^^JJx-x^)dl,  (2^> 

But  die  is  the  change  in  length  of  the  cable  due  to  a  change  in  unit  stress  in  it  of  df^,  or 


whence 

df. 


Vl^  /,  \    .  x\ 


E (30) 


This  is  the  change  of  sag,  or  the  deflection,  of  any  point  in  the  cable  {xy)  due  to  a  change  of 
unit  stress  in  the  cable  of  df,. 

The  deflection  of  B  due  to  the  stretch  of  the  hanger  is 


„      ,  dfh  dfi, 

dk    4  ^  =  (^^  -  y)~^ 


The  total  deflection  of  B  due  to  the  cable  system  is  therefore  dy  -\-dlhy  as  given  by  eqs. 
(30)  and  (31). 

166,  Deflection  of  the  Stay  System  for  Full  Loads. — The  length  of  a  stay  in  terms 
of  V  and  x  is 

=     -  x\ 

whence 

*,  =  '^rf/.  =  ^'f  =  (!i±i:")f  (3.) 

V   '      V  E      \    V    I E  ' 

This  is  the  deflection  of  B  due  to  a  stretch  of  the  stay  OB.  But  the  bottom  chord  also 
stretches  between  the  corresponding  stay  attachments  at  the  two  ends,  which  allows  of  a 
corresponding  diminution  in  x,  for  when  BC  increases,  AB  decreases  by  a  like  amount.  If  the 
truss  were  rigidly  fastened,  or  buttressed  at  A,\v&  might  assume  a  compression  in  AB ;  but 
although  it  is  anchored  down  at  A,  it  must  be  allowed  to  expand  and  contract  from  tempera- 
ture, and  hence  the  anchor-rods  permit  of  some  longitudinal  motion  here.  We  are  now 
solving  for  a  full  load,  and  hence  the  stays  will  pull  symmetrically  on  the  bottom  chord. 
When  one  end  only  is  loaded,  the  loaded  end  of  the  truss  is  pinned  fast  to  the  pier  by  the 
load,  while  the  opposite  end  is  swung  free.  The  stays  at  the  unloaded  end  also  are  slack  since 
the  truss  is  here  deflected  upwards,  and  hence  they  cannot  counteract  the  horizontal  pull  in 
the  stays  at  the  loaded  end.  In  this  case  the  horizontal  component  of  the  stress  in  the  stays 
is  taken  by  the  bottom  chord  in  compression  from  the  foot  of  the  stay  to  the  pier  at  that  end, 
provided  it  is  held  to  place  here.    This  case  will  be  discussed  below. 

If  we  assume,  for  full  loads,  that  about  five-eighths  of  the  length  of  the  bottom  chord  is  in  uni- 
form  tension,  from  the  stays,  of  dft  per  square  inch,  the  equations  will  be  simplified,  and  with  no 


174 


MODERN  FRAMED  STRUCTURES. 


appreciable  error.    The  stretch  of  bottom  chord  will  be,  therefore,  ^Z-^^,  and  since  this  will 

be  the  measure  of  the  dimumtion  of  x,  we  may  write 


But  from  the  triangle  A  OB  we  have  v'  — 


i6  e; 

jr",  whence 


dvt  =  —  -dx 


(33) 


S/x  ^ 
V         i6v'  Et 

This  is  the  deflection  of  j5  due  to  the  stretch  of  the  lower  chord. 

The  total  deflection  of  B  in  the  stay  system  is  therefore  dv,  4-  ,  as  given  in  eqs.  (32)  and 
(33)'  "ow  prepared  to  compare  the  deflections  of  the  two  systems. 

167.  Comparison  of  Deflections  in  the  Cable  and  Truss  Systems  for  Full  Load. — 

By  giving  numerical  values  to  x  and  to  —  in  eqs.  (30),  (31),  (32),  and  (33),  we  can  obtain 

numerical  coefiicients  for  the  expressions  /-^,  /^',         and  these  being  the  values  of 

^  E     E      E  10  Et  ^ 

the  stretch  from  full  live  loads  in  the  cable,  the  hangers,  the  stays,  and  the  lower  truss  chord, 

respectively.    In  the  following  table  such  coefficients  are  given  for  x  =.  J^,  -J,  ^3^,  and  ^  of  /, 

II 

and  also  for  —  =  12  and  —  =  16,  these  being  about  the  limiting  values  of  the  ratios  of  length 
to  versed  sine  for  good  practice  for  long  spans. 
TABLE  OF  DEFLECTION  COEFFICIENTS  OF  CABLE  AND  STAY  SYSTEMS  FOR  FULL  LOADS. 


Items. 

d/c       d/h       df,              I  d/t 
Coefficients  of /^r,              i-^,  and  —, -p^  for  dy,  d/i,  dv,,  ani  tit',. 

£        .c'                      S     £t                '     s'  c 

X  = 

X  =  y. 

X  =  A/. 

x  =  i/. 

X  =  11. 

X  = 

x  =  il. 

0.54 
0  .06 
0.  13 
0.48 

1 .01 
0.05 
0. 27 
0.94 

1 .42 
0.03 
0.50 
1.40 

1 .72 
0.02 
0.83 
1. 81 

0. 71 

0.04 
0. 12 
0.62 

1-33 
0.03 
0.31 
125 

1.89 
0.02 
0.62 

1.87 

2.27 

O.OI 
I  .06 
2.50 

Before  we  can  use  this  table  we  must  decide  on  what  the  changes  in  unit  stress  will  be  in 
the  several  parts.  The  dead  load  is  supposed  to  rest  wholly  on  the  cable  and  hangers,  the 
stays  and  truss  being  without  stress.    The  dead  load  may  also  be  taken  as  equal  to  one  half 


TV 


the  live  load,  or  ^  =  — .    Hence  the  change  of  stress  in  the  hangers  and  cable  for  full  live 

load  will  be  two-thirds  of  the  total  stress,  or  say  24,000  lbs.  per  square  inch.  The  stays  may  be 
stressed  to  40,000  lbs.  per  square  inch,  all  of  which  is  due  to  live  load.  The  bottom  chord  of  the 
truss,  if  of  structural  steel,  may  have  a  tensile  stress  of,  say,  io,ooo  lbs.  per  square  inch  added 
to  it  for  live  load  from  the  stays,  since  there  is  then  no  bending  moment  in  the  truss  to  speak 
of.  If  the  truss  is  of  timber,  then  it  may  have  500  lbs.  per  square  inch  added  to  the  total 
lower  chord  section,  thus  making  the  stretch  of  this  member  the  same  in  both  cases,  since 


10,000 


500 


28,000,000  1,400,000 


=  0.00036 


*  The  truss  may  be  of  timber,  and  hence  it  is  necessary  to  use  a  different  modulus  of  elasticity  here. 


SUSPENSION-BRIDGES, 


I7S 


If  we  now  put 


E 
E 

and  take  / : 


dfc  _  24,000 
E  ~  28,000,000 


=  0.00086, 


40,000 
28,000,000 

10,000 


=  0.00143, 

500 


2  8 ,000,000  1 ,400,000 
1000  feet, 


=  0.00036, 


(34) 


we  obtain  the  following  table  of  actual  deflections  : 


TABLE  OF  DEFLECTIONS,  IN  FEET,  OF  STAY  AND  CABLE  SYSTEMS. 

Length  of  Span  =  1000  Feet. 


Items. 


Deflection  of  cable  =  dy  

Stretch  of  hanger  =  dlh  

Deflection  of  cable  system      dy  -\-  dli, 

Deflection  from  stay  =  dvs  

Deflection  from  truss  chord  =:  dvt. ... 
Deflection  of  stay  system  =  dvs-\-  dvt 
Ratio  of  cable  system  to  stjiy  system. 


Versed  sine  =  —  span  or  v  = 


ft. 
0.46 
O.  II 

0.57 

o.iS 
0.18 
0.36 
1. 61 


ft. 
0.86 
0.08 
0.94 
0.38 
0.34 
0.72 
1.30 


ft. 

I  .22 

0.  04 

1 .  26 
0.72 
0.50 
I  .22 
1.03 


ft. 
1.48 
0.02 
1.50 
1. 18 
0.66 
1.84 
0.82 


Versed  sine  -  —  span  or  »  =  — . 

16  16 


ft. 
0.61 
0.04 
0.65 
0.17 
0.22 

0.39 

1.66 


x  =  k/. 


ft. 

1. 14 
0.03 
1. 17 
0.44 
0.45 
0.89 
1. 31 


ft. 

1 .63 
0.02 

1.65 

0.89 
0.67 

1.56 

1 .06 


The  last  line  in  this  table  gives  the  ratios  of  the  deflections  of  the  two  systems  /or  the 
assumed  unit  stresses.  But  since  these  deflections  must  be  equal  in  all  cases,  we  may  reason 
back  to  the  actual  stresses  and  conclude  that  the  last  liiie  in  this  table  shoivs  also  the  ratio  of 
actzial  to  the  assumed  stresses  in  the  stays,  for  the  stresses  in  the  cable  system  rcviaining  constant. 
That  is,  if  the  change  in  unit  stre.ss  in  the  cable  system  for  live  load  is  to  be  24,000  lbs.  per 
square  inch,  then  the  stay  attached  y'^/  from  the  pier  will  be  stressed  to  1.6  X  40,000  lbs.  per 
square  inch,  or  64,000  lbs.  per  square  inch  ;  the  one  attached  \l  from  the  pier  will  be  stressed 
to  1.3  X  40,000  lbs.  —  52,000  lbs.  per  square  inch  ;  the  one  attached  -^^l  out  will  be  stressed 
to  1.05  X  40,000  =  42,000  lbs.  per  square  inch  ;  while  the  one  attached  \l  from  the  pier  will 
be  stressed  to  but  0.8  X  40,000  =  32,000  lbs.  per  square  inch. 

These  results  are  wholly  independent  of  the  absolute  or  relative  sizes  of  the  stays  and  cable, 
and  of  the  loads  to  be  carried.*  Therefore  if  these  ratios  of  variation  of  working  stresses  can 
be  allowed,  the  stays  may  be  made  of  any  desired  size,  independent  of  all  other  dimensions. 
The  assumed  variation  of  unit  stress  in  the  cable  of  24,000  lbs.  per  square  inch  due  to  live- 
load  only  is  very  large.  If  this  be  reduced,  the  stresses  in  the  stays  will  be  reduced  accord- 
ingly. Since  the  stays  are  no  part  of  the  main  system,  it  is  thought  the  stresses  in  those  stays 
next  to  the  piers  may  very  well  be  from  to  |f  of  the  total  working  .stress  in  the  main  cable. 
When  the  stays  are  attached  farther  out  from  the  piers  than  -^^  of  the  span,  they  will  of  neces- 
sity have  a  less  unit  stress  in  them  than  obtains  for  the  cable,  whatever  their  size. 

168.  Action  of  the  Stays  under  Partial  Load- — If  the  live  load  cover  only  one  half  the 
span,  for  instance,  then  the  truss  is  supposed  to  distribute  this  load  evenly  upon  the  cable,  and 
is  therefore  deflected  upward  somewhat  on  the  unloaded  end.  This  causes  the  stays  to  be 
slack  at  that  end,  and  hence  they  cannot  exert  the  necessary  horizontal  pull  upon  the  truss  to 
balance  that  of  the  stays  at  the  loaded  end.    In  this  case  the  stays  act  exactly  like  the  sus- 


*  This  is  very  nearly  true  for  all  practicable  sizes  of  stays. 


176 


MODERN  FRAMED  STRUCTURES. 


pension  members  in  a  cantilever  bridge,  and  the  horizontal  components  of  the  stresses  in  them 
must  be  resisted  by  a  compressive  stress  in  the  bottom  chord  of  the  truss  back  to  the  pier  at 
the  loaded  end.  To  prevent  the  truss  from  sliding  back  over  the  pier  it  should  have  an  abut- 
ting resistance  so  arranged  as  to  allow  of  the  necessary  expansion  but  no  more.  It  can  then 
come  to  a  solid  bearing  and  the  stays  can  come  into  action.  Since  the  cable  system  does  not 
deflect  at  any  point  under  a  partial  load  as  much  as  it  does  under  a  full  load,  the  stays  will  be 
less  strained  under  the  load  here  taken  than  under  the  full  loads  already  discussed,  and  hence 
this  question  needs  no  further  consideration. 

169.  Stresses  in  Members  when  Stays  are  Used. — Since  the  maximum  unit  stresses  in 
the  stays  are  independent  of  their  size,  so  long  as  they  do  not  carry  the  whole  live  and  dead 
load  on  the  parts  reached  by  them,  they  can  be  made  of  any  desired  size.  If  they  are  all  of 
the  same  size,  they  carry  diminishing  loads,  as  their  points  of  attachment  are  farther  from  the 
piers ;  or  if  they  are  intended  to  carry  equal  vertical  loads  then  they  must  increase  in  size  as 
the  secants  of  their  angles  with  the  vertical. 

Let  us  suppose  they  are  intended  to  carry  one  half  the  live  load  out  to  the  quarter  points. 
Then  if  /  =  total  live  load  per  lineal  foot  of  span, 

w=    "    dead  "     "      "       "    "  " 

d  =  spacing  between  hangers, 

a=      "  "       stay  attachments, 

P=  vertical  load  on  one  stay, 

J  =  angle  stay  makes  with  the  vertical, 


pa 

we  have  P  =■  —  and 
2 


Stress  in  Stay  =.  ^  sec /*  •   .   •    .    .  (35) 


pi  pi 

Since  the  total  load  carried  by  the  stays  is  supposed  to  be  j  •-  =  — ,  the  maximum  load 
on  each  cable  system  is  \{w  -\- 1/)/.    Hence  the 

Maximum  Load  on  each  Hanger  =  -{^v  -\-  ^pj  (36) 

Also,  from  eq.  (5), 

Maximum  Stress  in  Cable         =  -{^  -\-^p]\/ 1  +  (~)  •     •    •   •  (37) 

The  total  tensile  stress  in  the  lower  chord  of  the  truss  for  a  full  load  should  also  be  com- 
puted. For  this  loading  there  is  considerable  bending  moment  in  the  truss,  at  the  centre,  when 
stays  are  employed.  If  the  stays  carry  one  half  the  live  load  on  the  end  portions  and  reach 
to  the  quarter  points,  there  would  then  be  a  uniform  upward  pull  on  each  truss  from  the  cable 
of  I/  per  lineal  foot ;  a  uniform  downward  load  on  the  end  sections  of  per  lineal  foot,  and  a 
uniform  load  on  the  middle  portion  of  per  lineal  foot,  with  end  reactions  of  zero,  under  our 
assumptions.    These  loads  would  develop  a  bending  moment  in  the  truss  at  the  centre  of  the 

soan  of  —  .    When  the  stays  are  attached  to  the  lower  chord  the  sum  of  the  horizontal 
^  128 

components  of  the  stresses  in  all  of  them  is  carried  by  this  chord.    The  horizontal  component 

pa  .   pa  IT 

of  the  stress  in  any  one  stay  which  carries  a  vertical  load  of  — -  is  —  tan j.    Hence  we  have 

pa              pP  , 
Total  Tension  in  Bottom  Chord  =  2  —  tasy  +  ^^^^  (38) 


SUSPENSION-BRIDGES. 


177 


The  total  load  in  each  pier  is  the  sum  of  all  the  vertical  components  in  the  cables  and 
stays  leading  to  it  from  both  sides.  If  this  reaction  is  vertical,  which  it  always  should  be, 
then  the  horizontal  components  on  the  two  sides  are  equal. 

If  the  shore  cables  and  stays  are  all  symmetrical  with  those  on  the  suspension  side  (which 
is  likely  to  be  true  of  the  cable  but  not  of  the  stays),  then  the  vertical  components  on  the 
two  sides  are  also  equal.  Now  the  vertical  components  on  the  span  side  are  equal  to  the  load 
carried  ;  hence  we  have,  for  symmetrical  arrangement  on  span  and  shore  sides  of  pier, 


Total  Load  on  Pier         =  2^/  -]-         =  (/  + 


(39) 


and  if  each  pier  is  composed  of  two  towers,  then 

Total  Load  on  One  Tower  =  ^(/^  -|-  w)/.  (40) 

170.  The  Direction  and  Amount  of  Pull  on  the  Anchorage. — The  horizontal  compo- 
nent of  the  pull  on  the  anchorage  is  equal  to  that  at  the  centre 
of  the  span,  as  given  in  eq.  (2).    The  vertical  component  is 
equal  to  the  horizontal  component  into  tan  i,  where  i  is  the 
angle  the  cable  makes  at  the  anchorage  with  the  horizontal. 

Referring  to  Fig.  219,  and  taking  moments  about  C,  we 

have 


p^* — If 

I 


Fig.  219. 


sD 
2 


Hv-VD-  — =0,    or  V=H 


D 


sD^ 
2  ' 


(41) 


where  s  is  the  load  per  foot  horizontal  coming  upon  the  cable  on  the  shore  side,  including  its 
own  weight  (and  which  may  be  simply  its  own  weight) ;  H  is  the  horizontal  component  of 
the  pull  in  the  cable,  which  is  constant  from  one  anchorage  to  the  other  under  vertical  loads ; 
Vis  the  vertical  component  of  the  stress  in  the  cable  at  the  point  O,  which  will  be  taken  as 
the  origin  of  co-ordinates,  at  a  distance  D  from  the  pier,  and  at  a  height  v  below  the  top  of 
the  tower. 

Taking  moments  about  any  point  in  the  cable,  as  P,  remembering  that  there  is  never  any 
moment  in  the  cable  itself,  we  have 


Vx  =0, 

2 


or 


(42) 


Equating  (41)  and  (42)  we  find  the  equation  of  the  curve  of  the  cable  on  the  shore  side 


to  be 


s      ^    [ sD      v\  >  . 


To  find  the  angle  it  forms  with  the  horizontal  at  any  point  we  have 

dy     .      .       s        I  sD 


dx 


tan  t 


.  _   s        IsD  v\ 


Hence,  for  ;r  =  o,  or  at  the  anchorage,  distant  D  from  the  pier,  we  have 


V  sD 


(44) 


(45) 


Now     is  the  tangent  of  the  angle  a  straight  line  through  O  and  C  makes  with  the  hori- 

zontal,  and  when  the  load  on  the  cable  is  small  on  the  shore  side,  or  when  sD  is  very  small  as 
compared  with  2//,  then  the  cable  will  deviate  very  little  from  a  straight  line  on  the  shore  side 
of  the  pier. 


178 


MODERN  FRAMED  STRUCTURES. 


The  vertical  pull  on  the  anchorage  is,  therefore, 


(46) 


The  angle  of  the  cable  on  the  anchorage  side  at  the  tower  is  found  from  eq.  (44)  by 
making  x  =  D,  whence 


This  angle  should  be  such  as  to  produce  a  vertical  reaction  in  the  tower.  To  accomplish 
this  it  may  be  necessary  to  transfer  some  horizontal  components  of  stress  from  stays  to  cable 
on  the  saddle,  which  may  be  done  through  their  frictional  resistance  to  sliding  without  any 
special  means  of  attachment. 

Note. — While  the  above  analysis  shows  that  composite  structures  may  be  computed  and  stresses  found 
on  any  given  assumptions,  it  must  be  borne  in  mind  that  a  structure  like  a  wire  cable  suspension-bridge 
does  not  admit  of  very  nice  adjustment  of  initial  tension  amongst  its  members,  or  of  very  rigid  joint 
connections.  Therefore,  even  though  the  engineer  succeeds  in  obtaining  the  proper  distribution  of  loads 
on  the  completion  of  the  structure,  it  is  not  likely  to  hold  this  adjustment  any  great  length  of  time. 
Hence  it  is  common  practice  to  dimension  the  members  on  the  assumption  that  it  is  to  act  as  a  simple 
structure,  and  then  the  superfluous  systems  serve  as  so  much  additional  factor  of  safety. 


tan  angle  i  at  tower  = 


2H 


(47) 


SWING  BRIDGES. 


»79 


CHAPTER  XII. 
SWING  BRIDGES. 

171.  General  Formulae. — A  swing  bridge  when  closed  is  ordinarily  a  continuous  girder 
of  two  or  more  spans.  If  the  ends  of  the  arms  are  almost  touching  their  supports,  without 
producing  any  end  reactions  from  dead  load,  the  span  is  balanced  over  the  centre,  and  the 
continuity  of  the  two  arms  makes  the  bridge  simply  two  cantilevers  balancing  each  other. 
When  the  live  load  comes  on  one  arm,  that  arm  is  immediately  deflected  until  it  finds  an  end 
reaction.  The  unloaded  arm  rises  and  still  has  no  end  reaction.  If  the  live  load  covers  both 
arms  and  is  symmetrical  about  the  centre,  there  may  still  be  no  end  reactions.  The  bridge 
then  becomes  a  tipper,  and  the  condition  which  then  obtains  is  the  same  as  in  a  locomotive 
turn-table.  This  condition,  however,  will  not  be  discussed  here,  as  it  does  not  properly  come 
under  the  head  of  swing  bridges. 

In  order  to  simplify  the  problem,  and  at  the  same  time  make  it  general  in  its  application, 
let  us  assume  the  bridge  closed,  with  the  ends  raised  by  raising  their  supports ;  then  the 
reactions  become  functions  of  the  elasticity.  If  we  assume  the  supports  as  unyielding,  the 
distortions  of  the  bridge  proper  need  only  be  considered.  In  what  follows,  the  usual  assump- 
tion of  constant  moment  of  inertia  is  made.  This  assumption,  although  not  true,  is  an  error 
on  the  safe  side,  and  is  the  only  one  practicable  in  computing  the  stresses. 

For  a  beam  continuous  over  three  or  more  supports,  the  "  Theorem  of  Three  Moments  " 
applies  (eqs.  14  and  14a,  p.  137),  and  will  enable  us  to  find  the  moments  at  the  supports  ;  from 
these  the  reactions,  and  finally  the  stresses.  Since  all  loads  are  given  over  to  the  trusses  as 
panel  concentrations,  the  equation  {140)  for  concentrated  loads  will  be  used.    This  equation  is 

Mr  Jr       +  2MXlr  .  ,  "f  /.)  "f  +        =  -  ^Pr  -  -.(^  "  >^')  "  2Prl\{2k  -  ^k^  +  k\  (l) 

in  which  M^.^ ,  M^,  and  J/^^.,  are  the  moments  at  the  (r  —  i)th,  rth,  and  (r+  i)th  supports, 
respectively,  beginning  on  the  left  hand  ;/,._,  and  are  the  lengths  of  the  (r  —  i)th  and  rth 
spans;  Pr —  i  and  Pr  are  any  concentrated  loads  on  these  spans;  and  kl=  a  is  the  distance 
from  the  load  to  the  support  on  the  left. 

172.  For  a  Continuous  Girder  of  Two  Spans. — Fig.  220,  and  are  both  zero. 
Making  r  =  2  in  eq.  (i),  and  assuming  load  P^  on  one  arm  only,  we  have 


■■  a=^kli — *lp 

1' 

B 

c 

f 

"  

Fig.  220. 


PJ\{k-k^. 

PJ^ 


.  (2) 


i8o  MODERN  FRAMED  STRUCTURES. 

Passing  a  section  at  B,  and  equating  the  moments  of  the  forces  to  the  left  with  ^  as  a  centre, 
we  have  also 


M. 


Substituting  value  of      from  eq.  (2),  we  have 


Let  /,  =  «/, ;  then 


Similarly,  we  find 


Je.  =  ^-)|2(>+'»)-'K>+2-^2'»)+*'!  (3) 


^•=;(7T^i-^+^"i  <4> 

and  from  2  vert.  comp.  =  o  we  have 

then  substituting  values  for     and  J?, ,  we  have 

R^=^\k{l  +  2n)-k^\   .  (5) 

For  loads  in  span  /,  make  /,  =  «/,  and  make  a  =  kl^  =  distance  from  right-hand  support. 
Then  R^  becomes  ^3  and  R^  becomes  R^ . 

The  above  eqs.  (3),  (4),  and  (5)  are  all  that  are  necessary  in  computing  the  reactions  for 
any  continuous  girder  over  three  supports  with  unequal  spans. 

For  Equal  Spans,  l^  —  1^-=.  l\  .•.«=!. 

^.=^l4-5/^+>^'};  (6) 

R,  =  ^\lk-k^\;  (7) 


R.^-^\-k^k^\  (8) 

The  above  eqs.  (6),  (7),  and  (8)  are  all  that  are  necessary  in  computing  the  reactions  for  any 
continuous  girder  over  three  supports  with  equal  spans.  In  any  case,  it  is  not  necessary  to 
find  the  pier  moments  directly.  Having  once  the  reactions,  the  stresses  in  the  members 
are  easily  found  by  the  principles  of  statics. 


SWING  BRIDGES, 


i8i 


For  a  Continuous  Girder  over  Four  Supports  (Fig.  221),  a  condition  which  frequently 
obtains  in  swing  bridges,  with  mid-span  /,  unloaded,  we  have  from  eq.  (i),  making  r  =  2  and 
making  /,  =  /,  =  /, 


2Mil  +  /,)  +  MA  =  -  Pl\k  -ky,  (9) 

and  making  r  =  $, 

MJ,  +  2  +  /)  =  -  P^n2^  -  Ik'  +   (9«) 

II  a 
For  convenience  make  /,  =  -;     «  =  r- ;  then  k  =  -j-.   Make  3  +  8«  +  4«'  =  H\  then  from 

ft  *3  rlt^ 

eqs.  (9),  (97)  we  can  obtain  the  reactions  similarly  as  for  a  beam  continuous  over  three 
supports. 

The  various  steps  in  deducing  the  following  equations  will  not  be  given.    We  then  have 


{ff-^2n  +  2n*)k+{2n-\-2n*)k'\;  (10) 


^.  =  :^K3  +  io«  -f  gn*  +  2n')k  -  (2«  +  $«•  +  2«y»{; 


(") 


^.  =  ;J|-(»+3«'  +  2«')'^,+ («  +  3«'  +  2n^'|;  (12) 

K^?f\nk-n^\  (13) 


For  loads  in  span  l^,  a-=  distance  from  right-hand  support.  Then  becomes  R^ ,  becomes 
,  R,  becomes  R, ,  and  R^  becomes  R,. 

The  above  equations  (10),  (11),  (12),  and  (13)  are  all  that  are  necessary  in  computing  the 
reactions  for  any  continuous  girder  over  four  supports,  with  equal  end-spans,  and  mid-span 
unloaded, — conditions  usually  obtaining  in  a  swing  bridge. 

When  the  reactions  R,  or  R^  become  minus,  eqs.  (11)  or  (12),  and  the  girder  is  not  held 
down  sufficiently  by  its  own  weight  or  otherwise,  the  condition  of  the  beam  is  at  once  changed 
to  that  of  a  beam  continuous  over  three  supports.  As  swing  bridges  are  ordinarily  built,  it  is 
impracticable  to  hold  down  the  centre  supports.  This  subject  will  be  taken  up  again  further 
on  in  discussing  the  various  forms  of  swing  bridges  in  common  use. 

The  foregoing  equations  are  all  that  are  needed  in  computing  the  stresses  for  any  swing 
bridge. 


l82 


MODERN  FRAMED  STRUCTURES. 


CONSTANTS  FOR  REACTIONS,  P  =  1000  POUNDS,  FOR  BEAM  CONTINUOUS  OVER  THREE 

SUPPORTS,  WITH  TWO  EQUAL  SPANS. 
(For  loads  on  the  left  span  only.) 


No.  of  Equal  Panels 
in  each  Span. 

Values  for    =  ^■ 

'4 

^. 

2 

I  2 

-f  406.25 

+  687.50 

-  93-75 

1  3 

2  " 

592.6 
240.7 

SR,  =  +  833.3 

481.5 
851.9 

2Ri  =  +  1333-4 

74-1 
92.6 

2R3  -   -  166.7 

4 

1  -f-  4 

2  " 

3  " 

691.4 
406.3 
168.0 

=  +  1265.7 

367.2 

687.5 
914.0 

2/i*a  =  +  1968.7 

58.6 
93-8 
82.0 

2Rs  —  —  234.4 

5 

1  5 

2  ^' 

3  " 

4  " 

752.0 
516.0 
304.0 
128.0 

"SRx  =  +  1700.0 

296.0 
568.0 
792.0 

944-0 

—  +  2600.0 

48.0 
84.0 
96.0 
72.0 

2Ra  =  —  300.0 

6 

1  -S-  6 

2  " 

3  " 

4  " 

5  " 

792.8 
592.6 
406.25 

240.75 
103.0 

2Ri  =  -j-  2135.4 

247-7 

481.5 

687.5 

851.85 

960.7 

2Rt  =r  -f  3229.25 

40.5 

74-1 
93-75 
92.60 
63.70 

2Ri  =  -  364.65 

7 

1  7 

2  " 

3  " 

4  " 

5  " 

6  " 

822.2 
648.7 
484.0 

332.4 
198.2 
86.0 

'SRi  =  +2571.5 

212.8 
416.9 
603.5 
763-8 
889.3 
970.9 

SA",  =  +3857-2 

35-0 
65.6 

87.5 
96.2 

87-5 
56.9 

2R3  =  —  428.7 

8 

1  8 

2  " 

3  " 

4  " 

5  " 

6  " 

7  " 

844.3 
691.4 

544-5 
406.3 
279-8 
168.0 
73-7 

2Ri  =  +  3008.0 

186.5 
367-2 
536.1 
687.5 
815.4 
914.0 
977-6 

2Ri  =  +4484- 3 

30.  S 

58.6 
80.6 
93-8 
95-2 
82.0 
51-3 

2R3  —  "  492-3 

9 

1  -j-  9 

2  " 

3  " 

4  " 

5  " 

6  " 

7  ■" 

8  " 

001 .5 
725-0 
592.6 
466.4 

348.4 
240.7 

145-4 
64-5 

■2  A",  =  +  3444-5 

166.0 

327.8 

481-5 
622.8 
747.6 
851.9 
931-4 
982.2 

2Ri  =  -|-  5111.2 

52.8 
74-1 
89.2 
96.0 
92.6 
76.8 
46.7 

2R3  -  -  555.7 

lO 

1  -1-  10 

2  " 

3  " 

4  " 

5  " 

6  " 

7  " 

8  " 

9  " 

875-25 
752.0 

631-75 
516.0 
406.25 
304.0 
210.75 
128.0 
57-25 

•Si?,  -  +  3881.25 

149-5 
296.0 
436.50 
568.0 

687.5 
792.0 

878.5 
944-0 
985-5 

2A',=  +5737-5 

24-75 

48.0 

68.25 

84.0 

93-75 

96.0 

89.25 

72.0 

42.75 

2Ra  =-  618.75 

Check,  for  any  value  of  k,  {Rx  +     +  Ri)  —  P  —  1000  ;  also,  in  any  case  2Rx  +  2Ri  +  2Ri  =  2P  =  Siooo, 


SWING  BRIDGES. 


173.  In  Computing  the  Various  Reactions  for  each  truss  panel  point,  the  work  will 
be  very  much  simplified  by  first  computing  the  reactions  for  an  assumed  load  of,  say,  1000 
pounds  at  each  point  successively.  This  will  give  constants  for  reactions  for  each  panel  point, 
which  can  then  be  multiplied  by  the  ratio  of  the  true  load  to  1000  pounds  to  obtain  the  true 
reaction. 

a 

Again,  as  the  panels  in  each  arm  are  usually  of  the  same  length,  k  =  j  can  be  expressed 

by  the  fractions  ^,  ^,  \,  \,  etc. 

The  table  on  the  opposite  page  gives  the  constants  for  reactions  for  a  beam  continuous 
over  three  supports  of  two  equal  spans,  from  eqs.  (6),  (7),  and  (8)  for  values  ol  k  X  ^  from 
I  to  9  inclusive. 

The  dead  load  for  railway  swing  bridges,  in  general,  may  be  obtained  very  closely  by 
finding  first  the  weight  of  a  fixed  span  of  the  same  length,  and  from  this  deduct  the  weight 
of  the  turn-table  to  be  used. 

For  single-track  bridges,  where  the  live  load  is  nearly  3000  lbs.  per  lineal  foot. 

Total  weight  of  metal  =  5/'  +  35°^ '< 

"        "       "  turn-table  (generally)  =  /  X  400; 
"        "      "  metal  above  turn-table  =  5/'  —  50/. 


The  weight  per  foot  above  turn-table  may  be  taken  as  w  =  $1  —  50,  where  /  here  is  the 
total  length  of  the  two  spans  of  the  swing  bridge.  To  the  above  weight  add  400  lbs.  per  foot 
for  weight  of  track. 

For  double-track  bridges  add  from  70  to  90  per  cent  to  the  above  weights.  The  percent- 
age to  be  added  varies  indirectly  with  the  length  of  the  span.  The  live  load  for  highway 
bridges  is  the  same  as  given  in  Chap.  IV. 

The  live  load  for  railway  bridges  can  be  assumed  as  a  uniform  train  load  with  one  or  two 
engine  excesses;  unless  a  train  of  engines  is  specified,  when  an  equated  uniform  live  load  can 
be  used.  The  former  is  the  more  common,  and  will  be  used  in  all  the  subsequent  computa- 
tions. The  excess  used  in  each  case  is  the  difference  between  the  maximum  panel  concentration 
from  the  engine  and  the  uniform  train  load.  Where  two  engines  are  specified,  it  is  customary 
to  assume  the  excesses  to  be  equal  and  placed  at  the  nearest  panel  points. 

174.  Centre-bearing  Pivot;  Three  Supports.*  Figs.  220  and  222. — In  this,  the  entire 
weight  of  the  bridge  when  open  is  carried  by 
the  cross-beam  ee'  to  the  pivot  P.  When 
closed,  the  ends  of  the  bridge  are  raised  at  a, 
a',  i,  and  i'.  The  bridge  is  thus  a  continuous 
girder  of  two  spans  for  dead  load,  and  for  live 
load  so  long  as  the  end  reactions  are  positive,  a 
The  deflection  of  the  ends  under  dead  load  is 
very  accurately  computed  after  the  bridge  is 
designed,  by  the  method  explained  in  Chap. 
XV.  After  raising  the  ends,  wedges  are  in- 
serted at  e  and  e'  and  brought  to  a  firm  bearing,  thus  relieving  the  pivot  P  from  all  live  load 
except  the  panel  load  at  ee'.  The  ends  are  not  latched  down,  hence  there  can  be  no  down- 
ward or  negative  reaction. 

To  compensate  for  lack. of  proper  adjustments,  and  also  as  unequal  chord  temperatures 
affect  the  deflection  at  the  ends,  two  assumptions  will  be  made  in  computing  the  dead  load 
stresses:  1st,  the  ends  just  touching  with  no  positive  reactions;  and  2d,  the  end  reactions 
equal  to  those  of  a  continuous  girder  over  three  supports.    It  is  safe  to  assume  that  practically 

*  For  this  case  treated  with  moment  of  inertia  of  the  truss  variable  see  Art.  178a,  p.  196. 


E 

/ 1 

 c 

Li 

^ 

Tr,      ^        "  }'i    ^  1 

Fig.  222. 


MODERN  FRAMED  STRUCTURES. 


the  end  reactions  will  vary  between  the  limits  of  these  assumptions.  In  fact,  it  will  be  shown 
that  in  a  properly  designed  bridge  the  ends  should  at  all  times,  when  the  bridge  is  closed,  be 
raised  so  that  the  end  reactions  will  be  at  least  a  mean  between  those  existing  when  there  are 
no  positive  reactions,  and  when  the  end  reactions  are  equal  to  those  of  a  continuous  girder 
over  three  supports.  The  analysis  will  then  consider  the  following  cases : 
Case  I.    Bridge  swinging,  dead  load  only  acting. 

Case  II.  Bridge  closed,  ends  raised,  dead  load  only  acting,  continuous  over  three 
supports. 

Case  III.  Live  load  on  one  arm  only,  for  maximum  tension  in  lower  chord  and  maxi- 
mum compression  in  upper  chord,  also  maximum  web  stresses  from  end  towards  centre. 

Case  IV.  Live  load  on  both  arms.  With  a  uniform  live  load  on  one  arm,  a  train  comes 
on  the  other  arm  and  advances  until  the  whole  bridge  is  covered,  giving  maximum  tension  in 
the  upper  chord  and  maximum  compression  in  the  lower  chord,  also  maximum  web  stresses 
from  the  centre  to  the  end. 

The  stresses  to  be  used  will  be  the  largest  of  each  kind  obtained  by  combining  Cases  III 
and  IV  with  Cases  I  or  II. 

Case  I. — In  this  case,  the  two  arms  are  simply  cantilevers  balancing  each  other  over  the 
centre;  and  stresses  are  easily  determined  by  diagram,  or  otherwise,  beginning  at  the  end  of 
the  bridge  where  the  only  external  force  is  the  load  at  that  point. 

Case  II. — In  this  case,  the  bridge  acts  as  a  continuous  girder  over  three  supports.  As 
the  two  arms  are  equal,  the  constants  for  reactions  given  in  the  table,  Art.  172,  can  be  used. 

w 

Let  W  denote  the  dead  load  per  truss  panel  =  -  X  panel  length ;  then,  as  the  number  of 
panels  =  4, 

W  W 

=  (126S.7  -  234.4)  X  —  =  755b  X  ^°3i.3 
=  (1968.7  +  1968.7)  X    "   =    "    X  3937-4 

R^  =  R,  =   «    X  1031.3 

Check :  {R,-]- R,-\- R,)=  2W  =6  W. 


After  finding  R^ ,  the  stresses  are  easily  found  by  diagram  or  analytically. 

Case  III. — In  this  case,  we  wish  first  to  find  the  maximum  tension  in  the  lower  chord 
and  the  maximum  compression  in  the  upper  chord. 

*  Before  proceeding  further  it  will  be  necessary  to  investigate  the  law  of  the  variation  of 
the  moment  in  a  two-span  continuous  girder  due  to  a  load  at  any  point.    Let  ABC,  Fig.  223, 
^a-kl-APi  B  represent  such  a  girder.    By  eq.  (6)  the  reaction  at  A 


A 


r    due  to  a  load  P,  in  the  first  span  is 


h—  ■  P 

Fig.  223.  1      4  it  I  p 


*  The  following  method  of  finding  the  position  of  live  load  for  maximum  chord  stresses,  and  the  diagram,  Fig. 
224,  are  from  a  paper  by  Prof.  M.  A.  Howe  on  "  Maximum  Stresses  in  Draw  Bridges,"  published  in  the  Journal  oj 
the  Association  of  Engineering  Societies,  July,  l8g2. 


SWING  BRIDGES. 


a  positive  quantity  for  all  values  of  k  from  o  to  i.    The  moment  at  any  point  iVto  the  left  of 
,  at  a  distance  x  from  A,  is  equal  to  ^,  X  ^,  a  positive  moment.    For  a  point  N  to  the  right 
of     the  moment  is 

M  =  R,yi  X  -  P,X  {x  -  kl) 

^P,k{l-x^^=^  (14) 


This  moment  is  zero  when 


4 

or,  if  x^  is  that  value  of  x  for  which  the  moment  is  zero,  we  have 

X,  4 


/  ~5 


(15) 


To  the  left  of  this  point  of  zero  moment  the  moment  is  positive,  and  to  the  right  it  is  negative. 
For  loads  in  the  second  span,  is  negative  ;  hence  the  corresponding  moment  in  the  first 
span  is  at  all  points  negative. 

The  value  of  y  in  eq.(i5)  varies  fromo.8,  for  ^  =  o,  to  i  for     =  i,  and  hence  all  loads 

X 

in  the  first  span  cause  positive  moments  at  all  points  to  the  left  of  the  point  where  -j  =  0.8. 

X 

The  curve  of  eq.  (15)  is  given  in  Fig.  224,  the  scale  for  values  of  k  and  -j  being  the  same  as 

for  k  and      in  the  curves  for  R^. 

The  diagram.  Fig.  224,  contains  the  plotted  curves  ot  eqs.  (6)  and  (8),  giving  values  of 
R^  for  values  of  k  or  k'  between  o  and  i,  and  for  a  value  of  P^  or  /*,  equal  to  unity.  The  re- 
action, R^ ,  due  to  any  panel  load,  whether  from  live  or  dead  load,  is  then  found  by  reading 
of?  the  ordinate  to  the  proper  curve  for  the  value  of  k  or  k'  corresponding  to  the  panel  point 
in  question,  and  then  multiplying  by  the  panel  load.  The  total  reaction  due  to  any  load  is 
the  sum  of  the  partial  reactions  due  to  panel  loads  thus  found. 

It  is  seen  from  the  diagram  that  7?, ,  for  loads  on  the  second  span,  is  always  negative, 
having  a  maximum  negative  value  of  .096  when  k'  =  .577.  Also,  that  for  loads  on  the  first 
span  R^  is  always  positive,  varying  in  value  from  i  to  o  as  ^  varies  from  o  to  i. 

Maximum  Positive  Moment,  or  Maximum  Compression  in  the  Upper  Chord  and  Maximum 
Tension  in  the  Lower. — It  has  just  been  shown  that  any  load  in  the  second  span  causes  nega- 
tive moments  at  all  points  in  the  first,  and  that  any  load  in  the  first  span  causes  positive 

X 

moments  at  all  points  in  the  first  span  to  the  left  of  the  point  where  -j  =  0.8 ;  hence,  for  a 

maximum  positive  moment  at  any  point  to  the  left  of  this  point,  the  second  span  should  con- 
tain  no  loads  and  the  first  span  should  be  fully  loaded. 

X 

For  a  centre  of  moments  to  the  right  of  the  point  where  y  =  0.8,  those  joints  in  the  first 

span  should  be  loaded  for  which  the  point  of  zero  moments  is  on  the  right  of  the  centre  of 
moments  taken.  As  many  different  loadings  are  required  as  there  are  centres  of  moments  in 
one  fifth  the  span,  usually  not  more  than  one.  The  point  over  the  centre  support  is  not  used 
as  a  centre  for  positive  moments,  as  the  greatest  positive  moment  there  is  zero. 


MODERN  FRAMED  STRUCTURES. 


It  will,  however,  be  shown  that  for  all  members  to  the  right  of  the  point  where  j  =  0.8, 

we  can  for  all  practical  purposes  assume  the  first  span  fully  loaded  ;  then  for  a  uniform  live 
load  one  position  is  all  that  is  necessary  to  find  the  maximum  compression  in  the  upper  chord 
and  maximum  tension  in  the  lower. 


Fig.  224. 


The  position  of  the  loads  having  been  found  in  any  case,  the  reaction  is  obtained  as 
previously  explained,  and  thence  the  stresses  in  the  members. 

We  next  want  to  find  the  maximum  web  stresses  from  the  end  towards  the  centre. 

Maximum  Posith>e  Shear,  or  Maximum  Tension  in  those  Diagonals  Inclining  Downward 
toward  the  Right. — The  positive  shear  in  any  panel  is  equal  to  7?, ,  minus  the  loads  between 
the  left  support  and  the  panel  in  question.    From  this  it  follows,  for  a  maximum  positive 


SWING  BRIDGES. 


187 


shear  in  any  panel,  that  there  should  be  no  loads  in  the  second  span,  for  any  load  in  the 
second  span  causes  a  negative  reaction  at  the  left ;  that  all  joints  in  the  first  span  to  the  right 
of  the  panel  in  question  should  be  loaded,  for  any  load  in  the  first  span  causes  a  positive 
reaction  at  the  left ;  and  that  no  joints  to  the  left  of  the  panel  should  be  loaded,  for  any  load 
in  the  first  span  causes  a  less  reaction  than  the  load  itself. 

The  proper  position  of  the  loads  and  the  corresponding  value  of  having  been  found 
for  any  member,  the  stress  in  the  member  is  perhaps  best  found  by  subtracting  from  01 
adding  to  the  shear  the  vertical  component  of  the  stress  in  the  upper  chord  member  cut,  thus 
getting  the  vertical  component  of  the  web  stress. 

The  stresses  in  the  verticals  corresponding  to  the  stresses  in  the  above  system  of  diagonals 
should  also  be  found.  The  loading  giving  the  maximum  stress  in  any  diagonal  gives  also  the 
maximum  stress  in  the  vertical  meeting  the  diagonal  at  the  upper  chord  point. 

Case  IV. — In  the  previous  case,  the  live  load  was  confined  to  one  span.  We  will  now 
assume  two  trains  on  the  bridge.  One  train,  which  covers  only  the  second  span,  is  a  uniform 
live  load.  The  other  train  headed  by  engines  is  just  coming  on  the  first  span  and  advances 
until  the  entire  bridge  is  covered. 

Maxhnum  Negative  Shear,  or  Maximum  Tension  in  those  Diagonals  Inclining  Downward 
toivard  the  Left. — The  negative  shear  is  equal  to  the  loads  to  the  left  of  the  panel  in  question, 
minus  the  reaction,  i?, .  For  reasons  similar  to  those  given  in  the  preceding  case,  the 
conditions  for  a  maximum  negative  shear  in  any  panel  are  :  the  second  span  should  be  fully 
loaded,  and  the  first  span  should  be  loaded  from  the  left  end  to  the  panel  in  question.  The 
corresponding  maximum  stresses  in  the  verticals  should  be  found  in  this  case  also;  and  where 
the  same  one  is  treated  in  this  and  the  preceding  case,  the  greater  of  the  two  stresses  is  to 
be  taken.    The  actual  stress  in  any  Vv'cb  member  is  found  as  in  the  previous  case. 

Maxiimivi  Negative  Moment,  or  Maximum  Tension  in  the  Upper  Chord  and  Compression 
in  the  Lower  Chord.- — The  second  span  should  be  fully  loaded,  for  all  centres  of  moments  in 

X 

the  first  span.    For  centres  of  moments  to  the  left  of  the  point  where  -j  =  0.8,  no  loads  should 

be  in  the  first  span  ;  and  for  centres  of  moments  to  the  right  of  this  point,  such  joints  should 
be  loaded  for  which  the  point  of  zero  moment  is  on  the  left  of  the  centre  of  moments  taken. 
These  positions  of  loads  follow  from  the  reasons  given  in  the  previous  case. 

As  previously  stated,  combine  Cases  III  and  IV  with  Case  I  or  II  to  obtain  the 
maximum  tension  or  compression  in  any  member.  It  might  here  be  argued,  that  when  the 
live  load  covers  the  first  span  only,  the  second  span  may  rise  until  the  right  end  is  lifted  from  its 
support.  This  happens  frequently  in  swing  bridges  where  the  ends  are  not  sufficiently  lifted, 
and  the  conditions  which  then  obtain  are  such  that  the  first  span  is  treated  as  an  independent 
span  for  live  load  stresses,  and  the  dead  load  stresses  are  the  same  as  for  Case  I.  In  no  case, 
however,  does  this  combination  give  any  greater  stresses  than  those  obtained  by  combining 
Cases  III  and  IV  with  Case  I  or  II. 

NUMERICAL  EXAMPLE.* 

Let  us  take  a  bridge  of  twelve  panels,  Fig.  225,  with  <f  =  20  ft. ;  /=  120  ft.;  bB  =  25  ft.,  and gG  =  35  ft.; 
length  of  upper  chord  member  =  V (20)'  +  (2)''  =  20.1  ft.    The  lengths  of  the  diagonals  are  as  follows : 

aB  =  Be  =  32.0  dE  =  36. 9 

=  Cd  =  33.6  eF  =  38.6 

cD=         35.2  /C  =  4o.3 

The  dead  load  above  turn-table  or  weight  per  lineal  foot  w  =  5/  —  50  +  400  =  1 550  lbs.  Dead  load  per 
truss  panel  =  ^^-^  x  20  —  15,500  lbs.  The  live  load  will  be  taken  at  3000  lbs.  per  lineal  foot,  headed  by  two 
engine  excesses  of  20,000  lbs.  each  placed  two  panels  apart. 


*  For  the  solution  of  this  case  with  moment  of  inertia  taken  as  variable  see  p.  196. 


i88 


MODERN  FRAMED  STRUCTURES. 


Live  load  per  truss  panel  =  30,000  lbs. ;  live  load  excess  per  truss  panel  =  10,000  lbs. 

Case  I.  Bridge  swinging,  dead  load  only  acting. — Taking  two-thirds  of  a  panel  dead  load  as  applied  at 
the  lower  chord  points  and  one-ihird  at  the  upper,  the  joint  loads  for  b,  c,  d,  e,  and /  are  each  equal  to  10,300 
lbs.,  and  for  B,  C,  D,  E,  and  /^are  each  equal  to  5200  lbs.  The  load  at  a  will  be  taken  at  one  half  of  a 
panel  load.    (The  joint  load  a  may  be  considerably  more  than  this  in  some  cases,  owing  to  the  weight  of 


Fig.  225. 


locking  gear,  etc.)  The  stress  diagram,  Fig.  226,  is  then  constructed  by  drawing  first  the  diagram  for  joint 
a,  where  the  only  external  force  acting  is  the  half  panel  load,  then  passing  to  B,  b,  C,  c,  etc.  The  diagonals 
Be  and  Cd  are  not  in  action  in  this  case,  and  so  are  omitted.  For  a  check,  the  stress  in  /g  is  by  moments 
equal  to  [7750  x  120  +  15,500  (1  +2  +  3+4+5)  x  20]  -7-  35  =  160,000  lbs.  compression. 

The  dead  load  stresses,  as  scaled  from  the  diagram,  Fig.  226,  are  given  in  the  second  column  of  the 
table  of  stresses. 

Case  II.  Bridge  closed,  dead  load  only  acting,  contimious  over  three  supports. — In  this  case,  the  loads 
at  a  and  n  need  not  be  considered,  as  they  are  carried  directly  to  the  supports.  The  other  joints  are  loaded 
as  in  the  previous  case.  Using  the  constants  for  reactions.  Art.  173,  we  have  for  this  span  Ri  =  (2135.4  — 
364.65)  X  15.5  =  say  +  27,500.  lbs.  With  this  value  of  R\  and  with  the  joint  loads  distributed  as  in  the 
previous  case,  the  diagram,  Fag.  227,  is  drawn.  For  a  check,  the  stress  in  fg  may  be  found  by  moments. 
Thus,  L  -V?^  ,  ,  ■ 

fg  =  127,500  X  120  —  15,500  X  (1+2  +  3+4  +  5)  x2o}-i- 35  =  38,600  lbs. 


Case  I.  Case  H. 

Fig.  226,  Fig.  227, 


The  stresses,  as  scaled  from  the  diagram.  Fig.  227,  are  given  in  the  third  column  of  the  table  of  stresses. 

Case  III.  Live  load  on  one  arm  only,  for  maxiiman  compression  in  upper  chord  and  tnaxifnum  tension  in 
Imuer  chord  ;  also  maximum  web  stresses  frotn  end  toward  centre. 

Maximum  Chord  Stresses.  (Case  III.)— First  position  of  live  load  :  engines  headed  toward  left;  30,000 
lbs.  at  A  and/,  with  engine  excess  10,000  lbs.  at and       Using  the  constants  for  reactions,  Art.  173 

we  have 

Ri  =  2135.4  X  30  +  (792.8  +  406.25)  X  10  =  say  +  76,100  lbs. 

The  chord  stresses  are  now  readily  found  by  moments ;  thus, 

ab  =  76,100  X  20  -i-  25  =  say  60,880  lbs.  tension. 

In  the  panel  be,  the  diagonal  Be  is  in  action,  since  both  live  and  dead  load  shears  in  this  panel  are 
largely  positive.    Hence  the  stress  in  be  is  equal  to  that  in  ab  =  60,880  lbs.  tension. 


SWING  BRIDGES. 


189 


The  centre  of  moments  for  BC  is  at  c. 

20      20  I 

BC  ={76,100  X  2  —  40,000  X  1}  X  —  X         =  83,500  lbs. 

In  the  panel  cd  we  will  assume  cD  as  acting,  then  find  its  stress,  and  if  the  result  is  a  tensile  "stress  our 
assumption  is  correct;  but  if  not,  then  must  be  in  action.  If  cD  is  in  tension,  then  the  chord  stress  CD 
must  be  greater  than  DE.    If  CD  is  less  than  or  equal  to  D£,  then  Cd  must  be  in  action. 

Assuming  <rZ>  in  action,  then 

20        20  I 

CD  =  BC  ={76,100  X  2  —  40,000  X  I }  X  —  X        =  83,500; 

>-  (  >  20  20.  I 

DE  =  {76,100  X  3  —  40,000  X  2  —  30,000  X  I }  X  —  X        =  82,040. 

As  CD  is  greater  than  DE,  then  cD  must  be  acting.  Also  from  Case  II  we  see  that  cD  is  in  tension  for  dead 
load;  hence  our  assumption  that  cD  is  in  action  is  correct.  Therefore  the  stress  in  CD  =  stress  in  BC  = 
83,500  lbs.   The  centre  of  moments  for  DE  and  cd  is  at  d. 

cd  =  {76,100  X  3  —  40,000  X  2  —  30,000  X  I }  X  ^  =  81,630  lbs.; 

20  I 

DE  =  {     "    X  3  —    "     X  2  —     "     X  I }  X  "  X         =  82,040  lbs. 

The  stresses  in  de  and  EF  are  found  in  like  manner. 

For  the  members  ef  and  FG,  the  centre  of  moments  is  at /.    For  maximum  positive  moment  at  this 

point,  those  joints  should  be  loaded  for  which  the  value  of  ^  is  greater  than  ^.    From  the  diagram.  Fig. 

224,  we  see  that  ^  >  |"  for  values  of  k  greater  than  0.44  ;  hence,  for  point /,  the  joints  to  the  right  of  the 

point  where  k  =  0.44  should  be  loaded.  This  includes  joints  d,  e,  and  /.  If  the  bridge  under  consideration 
had  only  five  or  less  equal  panels  in  each  arm,  this  special  position  of  the  live  load  would  not  have  been 
necessary  ;  and  it  will  be  shown  that  for  all  practical  purposes  this  disposition  of  the  live  load  can  be 
omitted  in  any  case  without  altering  the  required  sectional  area  in  any  member.  This  is  especially  true 
when  the  live  load  is  headed  by  heavy  engine  excesses.  However,  for  this  span  we  will  find  the  stresses  for 
ef  and  FG  when  only  d,  e,  and  /are  loaded,  and  compare  them  with  the  stresses  obtained  in  these  members 
from  the  second  position  of  the  live  load.  Loading  d,  e,  and  /with  30,000  lbs.  at  d,  e,  and  /and  10,000  lbs. 
at  d  and /,  we  find 

Ki  =  {(406.25  +  240.75  +  103)  X  30  +  (406.25  +  103)  X  10}  =  +  27,590  lbs.; 

ef  =  {  27,590  X  5  —  (40,000  X  2  +  30,000  ^       ^  ^  =  —  16,940  " 

FG  =  {     "      X  5  -  (   "    X  2  X    "      X  I)}  X  "  X  =  +  17,020  « 

Second  position  of  live  load;  engines  headed  toward  right;  30,000  lbs.  at f ,  </,  ^,  and/ with  engine 
excess,  10,000  lbs.  at  d  and /. 

y?,  =  2135,4  X  30  +  (406.25  +  103)  X  10  =  +  69,150  lbs.; 

ef=  {69,150  X  5  —  (30,000  X  10  +  10,000  X  2)1  X  —  =  —  15,600  " 

20  I 

FG  =  {    "     X  5  —  (    "     X  10  +  10,000  X  2){  X  "  X         =  +  15,680  " 

Furthermore,  these  stresses  when  combined  with  the  dead  load  stresses.  Case  II,  are  again  reduced,  leaving 
the  combined  stresses  very  small.  For  instance,  ef  becomes  —  15,600  +  12,700  =  —  2900  lbs.;  and  when 
joints  d,  e,  and  /only  are  loaded,  ^becomes  —  16,940  4-  12,700  =  —  4240  lbs. 


IQO  MODERN  FRAMED  STRUCTURES. 

Now,  as  a  matter  of  fact,  the  compression  in  ef  under  Case  IV  when  combined  with  Case  I,  really 
determines  the  sectional  area  of  the  member,  and  practically  it  makes  no  difference  whether  we  take  the 
tension  in  ef  at  2900  lbs.  or  4240  lbs. 

Maximum  Web  Stresses.  (Case  III.) — Beginning  at  the  left  hand  : — Maximum  compression  in  aB, 
30,000  lbs.  at  b,  c,  d,  e,  and /,  with  10,000  lbs.  excesses  at  b  and  d. 

Ri  =  2135.4  X  30  +  (792.8  +  406.25)  X  10  =  say  +  76,100  lbs.; 

aB  =  76,100  X  If  =  -I-  97,400  lbs. ; 

Bb  =  max,  panel  concentration  =  —  40,000  lbs. 

Maximum  tension  in  Be.    30,000  lbs.  at  c,  d,  e,  and  /,  with  10,000  lbs.  excesses  at  c  and  e. 

Ri  =  1(2135.4  —  792.8)  X  30  +  (592.6  +  240.75)  X  loj  =  +  48,610  lbs. 

Ri  is  the  shear  in  panel  be  when  b  is  not  /oaded.  This  shear  minus  the  vert.  comp.  in  BC  is  the  vert.  comp. 
in  Be. 

(  20. 1       2    )  32 

Be=  148,610  —  48,610x2  X  X  >  X  —=  —  53,020  lbs. 

( ^  ^  27       20. 1  )  25 

Maximum  tension  in  Cd.  For  dead  load  cD  has  a  tension  of  9000  lbs.,  but  the  end  lift  may  be  excessive 
and  so  reduce  the  shear  in  this  panel  to  zero  for  dead  load,  thus  giving  the  maximum  tension  to  Cd,  as  here 
found.    Then  with  30,000  lbs.  at  d,  e,  and  /,  and  10,000  lbs.  at  d  and /, 

Rt  =1(406.25  +  240.75  +  103)  X  30  +  (406.25  +  103)  X  io|  =  +  27,590  lbs. 

Any  stress  in  Cd  must  be  accompanied  by  a  corresponding  difference  between  the  chord  stresses  BC 
and  CD,  and  their  difference  is  a  measure  of  the  stress  in  Cd. 

r.  .  20.  I 

Stress  m  CD  =  (27,590  x  3)  x  — -; 

29 

20.1 

"     "  BC—{    •'     X  2)  X  . ; 

27 

"  Cd  =  (CD  —  BC)  X 

20.1 

Therefore,  we  may  write  at  once  : 

Stress  in  Cd  =  27,590 (/^  —  j\)  x  33.6  =  —  27,790  lbs.,  and 

29 

"     "  Ce  =  stress  m  Cd  x  — -,  or 

33-6 

«     "  Ce  =  27,590  (2^  —  /^)  X  29  =  ■+  23,990  lbs. 

Case  IV.  Uniform  live  load  covering  the  second  span. — A  train  comes  on  the  first  span  and  advances 
until  the  whole  bridge  is  covered. 

Maximum  Web  Stresses.  (Case  IV.) — Beginning  at  left  hand  : — Maximum  tension  in  aB  and  maximum 
compression  in  bB.    40,000  lbs.  at  a,  second  span  fully  loaded. 

It  is  here  assumed  that  the  load  in  the  second  span  lifts  the  end  at  a,  owing  to  temperature  or  lack  of 
rtdjustment  in  raising  the  ends;  then  any  load  which  comes  on  at  a  holds  the  end  down. 

R\  for  load  in  second  span  =  —  364.65  x  30  =  —  10,940  lbs. 
Stress  in  aB  =  10,940  x  |f  =  —  14,000  lbs. 
"     "  bB  =    "     X  II  =  +  10,060  " 


SWING  BRIDGES. 


191 


Maximum  tension  in  bC.    40,000  lbs.  at  b,  second  span  fully  loaded, 

R\  for  load  in  second  span  =  as  before  —  10,940  lbs.; 
Rx   "    "    at  i5  =  792.8  X  40  =  + 31,710  lbs.; 
2J?i  =  +  31,710  —  10,940  =  +  20,770  lbs. 
Shear  in  panel  be  =  20,770  —  40,000  =  —  19,230  lbs. 
riiis  shear  plus  the  vert.  comp.  in  BC,  is  the  vert.  comp.  in  bC. 

Stress  in  bC 
«  "  Cc 
«     "  Cc 

For  the  maximum  tension  in  cD  and  maximum  compression  in  Dd.  30,000  lbs.  at  b  and  40,000  at  c, 
second  span  fully  loaded.  In  this  manner  we  proceed  until  the  centre  is  reached,  the  second  span  remam- 
ing  fully  loaded  under  all  conditions. 

Maxiinwn  Chord  Stresses.  (Case  IV.) — As  before,  tlie  second  span  is  fully  loaded  with  uniform  live 
load,  and  the  first  span  is  unloaded  for  all  centres  of  moments  except  at /  and  g.  However,  if  we  assume  that 
the  load  in  the  second  span  lifts  the  end  at  a,  as  previously  explained,  then  any  load  which  comes  on  at  a 


CHORD  STRESSES. 


Dead  Load. 

Live  Load. 

Total. 

Member. 

Case  I. 

Case  II. 

Case  III. 

Case  IV. 

+ 

BC 

—  6360 

4-  29700 

-  83500 

—  S800 

1 13200 

I5160 

CD 

—  23300 

-f  29700 

-  83500 

—  16270 

1 13200 

39570 

DE 

—  49300 

-j-  24400 

-  82040 

—  22750 

106440 

72050 

EF 

—  82700 

4-  9500 

- 

-  61 190 

—  28370 

70690 

II 1070 

FG 

—  I2I00O 

—  12700 

- 

-  15680 

—  34860 

2980 

155860 

ab 

+  6360 

—  22300 

-  60880 

+  8750 

15110 

83180 

be 

+  23300 

—  22300 

-  60880 

+  16190 

39490 

83180 

cd 

+  49300 

—  24400 

-  81630 

-j-  22640 

71940 

106030 

de 

+    8 1  goo 

—  9500 

-  60890 

-|-  28230 

IIOI30 

70390 

-j-  120800 

-4-  12700 

-  15600 

+  34690 

155490 

2900 

fg 

-j-  160000 

-j-  38600 

000 

4-  80360 

240360 

000 

\  20.  I  2  33.6  ^  ,, 

\  19,230  +  20,770  X          X          \  X        =  —  26,000  lbs.  ; 

j   ^  25       20. 1  i  27 


25 

stress  in  bC  x  — r;  therefore, 
33-6 


I  20. 1       2    )  25 

=  j  1 9,230  + 20,770 X  —  X  ^  [  X  ^  =  +  '9.340  lbs. 


WEB  STRESSES. 


Dead  Load. 

Live 

Load. 

Total. 

Member. 

Case  I. 

Case  11. 

Case  III. 

Case  IV. 

+ 

aB 

—  lOOOO 

+  35500 

+  97400 

—  14000 

132900 

24000 

bC 

—  28600 

000 

not  max. 

—  26600 

000 

54600 

cD 

—  46100 

—  9000 

-  44480 

000 

90580 

dE 

—  61000 

—  27300 

<t  14 

—  70140 

000 

131 140 

eF 

—  74200 

—  43200 

<t  <l 

—  98020 

000 

172220 

fG 

—  86900 

-  58300 

iC  H 

—  126500 

000 

213400 

Be 

000 

—  10700 

—  53020 

not  max. 

000 

63720 

Cd 

000 

000 

-  27790 

000 

27790 

Bb 

+  12700 

—  10600 

—  40000 

-f-  10060 

22760 

50600 

Cc 

-j-  27000 

+  5300 

+  23990 

not  max. 

50990 

000 

Dd 

4-  40500 

4-  12700 

not  max. 

+  34110 

74610 

000 

Ee 

-f  53000 

-j-  25900 

ft     1 1 

+  55100 

I0810O 

000 

Ff 

-j-  65200 

+  40300 

i<  *i 

+  78750 

143950 

000 

Gg 

-j-  180000 

4"  108600 

«•  n 

-j-  210250 

390250 

000 

192 


MODERN  FRAMED  STRUCTURES. 


holds  the  end  down.  Then  for  all  centres  of  moments  in  the  first  span,  except  at  /  and^,  we  will  assume  a 
load  at  a. 

R\  for  load  in  second  span  =  as  before  —  10,940  lbs. 

The  load  at  a  is  the  max.  panel  concen.  =  40,000  lbs.  Therefore  the  end  tends  to  lift  by  the  amount  of  the 
negative  reaction  from  load  in  the  second  span,  or  i?i=  —  10,940  lbs.  This  value  of  Ri  is  used  for  all  centres 
of  moments  from  ^  to  <?  inclusive. 

For  centre  of  moments  at  /,  those  joints  should  be  loaded  for  which  the  point  of  zero  moment  lies  to 
the  left  of  f,  i.e.,  joints  b  and  c.    With  30,000  lbs.  at  b  and  40,000  lbs.  at  c,  we  have 

R\  =  1 792. 8  X  30  +  592.6  X  4o|  =  +  47,490  lbs.; 
Rx  from  load  in  second  span     =  —  10,940  " 
2^,  =  +  36,550  " 

The  stresses  in  ef  and  FG  are  then  found  with  this  value  of  i?,  =  +  36,550  lbs. 

For  centres  of  moments  at^  or  Gthe  entire  bridge  must  be  loaded  with  the  engine  excesses  at  ^and /. 

Rx  from  load  in  first  span     =  +  69,150  lbs.; 
/?, second  span  =  —  10,940  " 
=69,150—  10,940  =  +  58,210  " 

Stress  iny^  =  1 58,210  x  6  —  (30,000  x  15  +  10000x4)}  x     =  +80,220  lbs. 

The  maximum  stresses  for  each  case  are  entered  in  the  table  of  stresses.  The  total  stresses  are  entered 
in  the  last  two  columns  of  the  table. 

In  all  of  the  foregoing  analysis  and  computations,  we  have  assumed  two  conditions  which 
are  extreme  but  not  impossible.  First,  we  have  assumed  two  trains  on  the  bridge  at  the 
same  time — a  condition  which  may  or  may  not  occur  in  the  lifetime  of  a  structure ;  and  even 
then  the  positions  necessary  to  give  maximum  stresses,  as  previously  assumed,  may  not  occur. 
Again,  we  have  assumed  the  ends  insufficiently  raised,  so  as  to  permit  the  ends  to  "hammer," 
as  it  is  called.  The  latter  condition  obtains  frequently  in  swing  bridges  where  the  proper 
lifting  of  the  ends  is  neglected.  In  practice  it  is  customary  to  assume  only  one  train  on  the 
bridge.  Then,  in  Case  IV,  we  assume  the  train  headed  by  engines  coming  on  one  end  of  the 
bridge  and  advancing  until  the  whole  bridge  is  covered.  As  the  train  comes  on  the  first  span 
aijd  advances  we  then  obtain  a  condition  for  maximum  web  stresses,  or  maximum  negative 
shear  at  the  head  of  the  train.  When  the  web  members  near  the  centre  of  the  bridge  are 
reached,  we  must  then  permit  the  train  to  advance  until  the  whole  bridge  is  covered  to  obtain 
maximum  web  stresses.  This  position  of  the  live  load  will  also  give  maximum  chord  stresses 
near  the  centre.    We  see,  then,  that  for  all  practical  purposes  it  is  never  necessary  to  find  the 

distance  y^,  or  the  point  of  zero  moment  for  Cases  III  or  IV. 

It  is  also  customary  to  assume  that  the  ends  of  the  bridge  are  properly  raised  so  as  to 
preclude  the  possibility  of  their  "hammering,"  even  under  extreme  conditions  of  temperature 
when  the  train  comes  on  the  bridge.  In  any  event,  however,  the  conditions  for  dead  load 
under  Cases  I  and  II  must  be  assumed. 

The  cases  to  be  considered  should  then  be  stated  as  follows: 

Case  I.  Bridge  swinging,  dead  load  only  acting. 

Case  II.  Bridge  closed,  ends  raised,  dead  load  only  acting,  continuous  over  three  supports. 

Case  III.  Live  load  on  one  arm  only,  for  maximum  tension  in  lower  chord  and  maximum 
compression  in  upper  chord,  also  maximum  web  stresses  from  end  towards  centre. 

Case  IV.  Live  load  on  both  arms,  for  maximum  tension  in  upper  chord  and  maximum 
compression  in  lower  chord,  also  maximum  web  stresses  from  the  centre  towards  the  end. 

Cases  III  and  IV  to  be  combined  with  Case  I  or  II,  as  previously  explained. 


SWING  BRIDGES. 


193 


175.  Rim-bearing  Turn-table — Four  Supports.*— Fig.  228. — The  most  common 
method  of  swing-bridge  construction  provides  two  supports  at  the  centre,  thus  dividing  the 
truss  into  three  spans.  Tiie  bracing  of  the 
short  central  span  is  usually  arranged  as  in  Fig. 
228.  It  is  thus  capable  of  resisting  shear,  and 
the  continuity  of  the  truss  is  complete. 

With  some  writers  it  has  been  customary 
to  apply  to  this  form  of  truss  the  formula;  for  a 
continuous  beam  of  four  supports — formulae  based  on  the  assumption  of  uniform  moment  of 
jnerlia  and  on  a  neglect  of  deflection  due  to  shearing  stresses.  These  formulae  give  results 
closely  approximate  for  beams  and  long-span  trusses,  and  fairly  good  results  for  two-span 
swing-bridges;  but  their  application  to  trusses  of  such  short  spans  as  are  here  considered  leads 
to  very  erroneous  conclusions.  With  only  one  span  loaded  large  negative  reactions  are 
obtained  at  B  or  C,  reactions  much  greater  than  ever  really  occur,  and  which  greatly  exceed 
the  dead-load  positive  reactions.  To  furnish  these  negative  reactions  some  form  of  anchorage 
would  have  to  be  provided  at  B  and  C,  a  thing  quite  impracticable  in  a  swing-bridge. 

The  true  stresses  in  the  diagonals  EC  and  FB,  caused  by  partial  loads,  and  their  effect  on 
reactions,  may  be  estimated  in  the  following  manner:  Assume  first  that  these  diagonals  are 
removed  and  one  span,  as  AB,  loaded.  The  reactions  and  stresses  are  now  to  be  computed 
by  treating  the  bridge  as  a  two-span  bridge,  or,  more  accurately,  by  the  formula  of  Art.  176. 
Calculate  then  by  the  method  explained  in  Chap.  XV  the  deflection  of  the  y>o\u\.  E  vi  the  direc- 
tion CE.  Now  if  the  diagonal  CE  had  been  in  place  its  elongation  could  not  have  exceeded  this 
movement  of  E.  If  the  diagonal  be  small  its  elongation  may  be  assumed  equal  to  this  deflec- 
tion, and  the  stress  per  square  inch  computed  which  would  result  therefrom.  If  the  diagonal 
be  large  it  would  have  the  effect  of  reducing  somewhat  the  movement  of  E,  and  therefore 
would  have  a  smaller  stress  per  square  inch  than  a  small  diagonal.  To  the  total  stress  as  deter- 
mined above  should  be  added  the  initial  tension  due  to  adjustment.  The  reaction  at  C  will 
then  be  equal  to  the  stress  in  EC,  minus  the  vertical  component  of  the  diagonal  stress.  Com- 
putations for  two  structures  gave  in  one  case  a  stress  of  800  lbs.  per  sq.  in.  in  the  diagonal, 
and  in  the  other  case  1800  lbs.  per  sq.  in.,  the  diagonals  in  both  cases  being  very  small. 
Increasing  the  area  in  the  latter  case  to  12  sq.  in.,  the  size  actually  used,  reduced  the  stress 
per  square  inch  to  1000  lbs.,  thus  giving  a  total  stress  of  12,000  lbs.  The  stress  in  EC  was 
18,000  lbs.  and  the  resulting  reaction  therefore  about  8oDD  lbs.,  positive,  neglecting  initial 
tension.  The  formuhE  for  a  four-span  continuous  girder  would  give  a  negative  reaction 
of  about  200,000  lbs.  and  a  diagonal  stress  somewhat  greater.  The  diagonals  were  apparently 
designed  for  this  stress. 

The  above  discussion  indicates  that  the  effect  of  bracing  in  the  centre  span  is  small,  and 
in  the  computations  of  stresses  this  bracing  can  be  entirely  neglected  and  the  truss  treated  as 
as  in  Art.  176,  or  approximately  as  a  two-span  bridge.  Since  the  purpose  of  this  bracing  is 
merely  to  stiffen  the  structure  when  open,  small  members  should  be  used.  Large  ones  cause  a 
more  unequal  distribution  of  load  on  the  rollers,  and  should  be  avoided, 

176.  Rim-bearing  Turn-table ;  Four  Supports ;  Equal  Moments  at  the  Centre 
Supports. — Fig.  229. — In  this  the  rigid  frame  EEBC  supports  the  truss  by  means  of  the  short 

G    H  links  EG  and  EH.    The  portion  BC  of  the 


frame  is  a  part  of  the  lower  chord  of  the  bridge  ; 
in  other  respects  the  frame  may  be  considered 
a  part  of  the  pier.  There  being  no  diagonals 
in  the  panel  EEHG,  there  can  be  no  shear 


229  transmitted  across  this  panel,  and  the  moments 

at  //and  G  must  be  equal.  The  structure  may  then  be  considered  a  three-span  continuous 
girder,  in  which  the  moments  at  the  two  centre  supports  are  always  equal. 


*  This  article  changed  in  the  Sixth  Edition. 


194 


MODERN  FRAMED  STRUCTURES. 


Now  if  the  first  and  third  spans  be  each  loaded  Avith  a  single  load,  P,  placed  symmetri- 
cally with  reference  to  the  centre,  the  moments  at  the  second  and  third  supports  will  be  equal 
to  those  which  would  occur  in  an  ordinary  continuous  girder  of  three  spans,  if  loaded  in  the 
same  way,  for  in  the  latter  case  and  J/j  would  be  equal  by  symmetry  and  therefore  the 
shear  zero  in  the  second  span. 

Also  these  moments  and  are  now  each  produced  equally  by  the  two  loads  and 
,  hence  each  may  be  supposed  to  be  produced  wholly  by  one  half  the  sum  of  the  moment 
effects  of  the  two  loads  taken  separately.    Thus,  from  eqs.  (9)  and  (9<?),  p.  181,  we  have 

M._=.M,=  -^^^iPiK-K')^Pi2K-iK^-\.K^)\  (17) 

Now  if  the  load  P^  be  removed  from  the  third  span  we  shall  have  * 

^.=  -J^.(^-n  (.8) 

We  also  have  M^  —  R^y.  I  —  P{1  —  kl),  and  M^  —  R^y^l  —  M^,  whence,  by  solving  for  R^ 
and      and  substituting  for       its  value  in  eq.  (18),  we  have 


R 

and 


Using  eqs.  (19)  and  (20)  in  getting  the  reactions  due  to  the  various  panel  loads,  the 
analysis  is  otherwise  precisely  the  same  as  for  a  two-span  bridge,  Art.  174. 

The  two  links  GE  and  HF  are  to  be  treated  as  posts  of  the  web  system.  The  maximum 
stress  in  GH  is  equal  to  the  maximum^  in  LBCK.  The  stresses  in  EB  and  EC  are  equal 
respectively  to  those  in  GE  and  HE.  The  pieces  EE,  EC,  and  BE  receive  no  definite  stress; 
they  serve  merely  to  afford  stability  to  the  columns  EB  and  EC. 

The  analysis  of  the  case  where  light  diagonals  are  inserted  in  the  panel  GHBC  sufficient 
only  to  give  stability  to  the  bridge  when  open  is  the  same  as  for  the  preceding  case. 

177.  Rim-bearing  Turn-table  ;  Three  Supports. — In  this  the  single  link  GP  carries 
the  load  at  the  centre  to  the  rigid  frame  Pgh. 
The  length  of  the  panel  gh  is  made  equal  to  the 
width  of  the  truss  in  order  that  the  weight  upon 

the  turn-table  may  be  uniformly  distributed.   The   a  y    i    «    i    y         /*    <  "  " 

length  of  the  other  panels  in  the  lower  chord  is  Fig.  230. 

independent  of  gli.  The  analysis  is  precisely  the  same  as  for  a  simple  two-span  continuous 
girder. 

Example. — Find  the  maximum  stresses  in  all  the  members  of  the  truss  in  Fig.  230.  Length  ag  =  120  ft. ; 
g/i  =  18  ft. ;  Ao  =  1 20  ft. ;  height  at  B  and  N  —  24  ft.,  and  at  (7  =  36  ft.  Total  dead  load  =  1650  lbs.  per  foot. 
Take  for  the  live  load  the  equivalent  uniform  load  for  a  span  of  150  ft.    (See  Chap.  VI.) 

The  span  lengths  to  be  considered  are  120  +  9  =  129  ft.  each. 

*  This  method  of  arriving  at  Afi  for  a  single  load  is  an  expedient  resorted  to  here  to  avoid  a  recourse  to  ihe 
principle  of  least  work.  The  problem  is  indeterminate  by  the  ordinary  principles  of  statics.  For  an  outline  of  the 
rigid  method  see  an  article  by  Prof.  Merriman  in  Engi7iei:iing  News,  Sept.  5,  1895,  Vol.  XXXIV,  p.  150. 


SWING  BRIDGES. 
The  stresses  due  to  dead  load  when  the  bridge  is  open  are  found  by  diagram  as  before. 


195 


Joint. 


b 
c 

d 
e 

1 
z 

k 
I 

m 
n 


•155 
.310 
.465 
.620 
•775 
•775 
.620 
•  465 
.310 

•155 


Rx  for  P=x. 


620 

444 
284 
148 
079 
094 
og2 
070 
038 


When  the  bridge  is  closed  and  ends  raised  it  is  a  continuous  girder  of  two  spans  of  129  ft.  each,  under 
dead  or  live  loads.  The  various  values  of  k  or  k'  for  the  different  joints,  together  with  the  corresponding 
values  of  R\  as  found  from  the  diagram  Fig.  224,  are  given  here.  The  student  is  left  to  complete  the 
analysis. 

177a.  Rim-bearing  Turn-table  ;  Four  Supports. — Equal  Moments  at  the  Centre  Sup- 
ports;  also,  Equal  Loads  on  the  Turn-table  spaced  at  Equal  Distances. — Fig.  231. — In  this  the 
two  triangular  frames  BHC  and  CID  support  the  truss  by  means  of  the  short  links  FH  and 
GI.    The  portions  BC  and  CD  of  the  frames  are  a  part  of  the  lower  chord  of  the  bridge. 
The  point  C  is  common  to  both  frames.    The  inclinations  of  the  members  BH,  HC,  CI,  and 


Fig.  231. 

ID  is  such  that  under  maximum  loads  at  H  and  /  the  supports  B,  C,  and  D  are  equally  loaded. 
This  arrangement  brings  the  weight  on  the  turn-table  uniformly  distributed  over  six  points. 
These  points  can  be  spaced  equal  distances  apart  on  the  turn-table.  Under  all  conditions,  the 
moments  at  the  two  centre  supports  F  and  G  are  always  equal.  The  analysis  will  then  be  the 
same  as  in  Art.  176.  However,  in  all  cases  of  this  kind,  where  the  centre  span  is  short  as 
compared  with  the  side  spans,  it  is  customary  in  the  computations  to  assume  the  centre 
span  left  out.    The  bridge  then  becomes  a  swing  bridge  with  two  equal  spans,  as  in  Art.  174. 


Lift  Swing  Bridges. 

Various  devices  have  been  suggested  whereby  the  bridge  may  be  made  continuous  when 
being  opened  and  be  two  simple  spans  when  p 
closed.    Fig.  231  {a)  shows  a  form  in  which, 
when  the  bridge  is  to  be  swung,  the  supports 
at  B  and  C  are  lifted  far  enough  to  bring  the 
links  EF  and  EG  into  action  and  to  raise  the 
ends  A  and  D  from  their  supports.    All  the  weight  is  then  at  the  centre  and  the  bridge  is 
swung  on  a  centre-bearing  pivot.    When  closed,  and  the  supports  B  and  C  lowered,  the  links 
EF  and  EG  are  under  no  stress.    The  analysis  then  consists  in  finding  the  dead  load  stresses 


7B  c 


Fig.  231  {a). 


196 


MODERN  FRAMED  STRUCTURES. 


when  open,  as  in  the  other  forms,  and  the  dead  and  live  load  stresses  when  closed,  the  bridge 
then  consisting  of  two  simple  spans. 

178.  Wind  Stresses. — The  analysis  for  the  stresses  in  the  laterals,  forming  the  web 
systems  of  the  wind  trusses,  is  carried  on  precisely  as  for  the  vertical  or  main  trusses.  If  the 
bridge  to  be  considered  is  a  through  bridge,  the  top  lateral  wind  loads  must  be  considered  as 
being  transmitted  to  the  supports  through  bending  in  the  web  members.  The  routes  which 
they  select  in  any  case  will  be  such  that  the  internal  work  done  in  producing  strain  will  be  a 
minimum.    The  conditions  to  be  considered  are  then : 

Top  Laterals : 

Case   I.    Bridge  swinging.    Static  wind  pressure  only. 

*'  II.    Bridge  closed.    Ends  raised.    Static  wind  pressure  only. 

Bottom  Laterals : 

Case    I.    Bridge  swinging.    Static  wind  pressure  only. 

"    II.    Bridge  closed.    Ends  raised.    Static  wind  pressure  only. 
"  III.    Bridge  closed.    Live  load  on  one  arm  only. 
"  IV.    Bridge  closed.    Live  load  on  both  arms. 

Combine  Cases  III  and  IV  with  II,  or  consider  I  or  II  alone. 

For  Cases  I  and  II,  when  considered  alone,  it  is  customary  to  assume  wind  pressures  of 
30  and  50  lbs.  per  square  foot  respectively.  When  the  train  comes  on  the  bridge,  a  wind 
pressure  of  50  lbs.  would  overturn  the  cars;  so  that  for  Cases  II,  III,  and  IV,  when  combined 
as  previously  explained,  a  wind  pressure  of  30  lbs.  per  square  foot  is  assumed,  which  is  about 
the  maximum  wind  pressure  that  will  not  overturn  a  standard  box  car  when  fully  loaded.  It 
is  easy  to  see  that  in  some  bridges  Cases  I  and  II,  when  considered  alone,  might  give  a  maxi- 
mum in  some  of  the  members. 

The  chord  stresses,  resulting  from  wind  loads,  should  be  considered  where  they  increase 
the  chord  stresses  in  the  main  trusses. 

178a.  Accuracy  of  the  Ordinary  Formulae  for  Swing-bridges.* — On  p.  193  it  was  noted 
that  there  is  an  error  in  the  ordinary  formulae  due  to  assuming  a  constant  value  for  the 
moment  of  inertia  and  to  the  neglect  of  the  shearing  distortion,  and  it  was  shown  that  this 
error  is  very  great  in  the  case  there  discussed.  It  is  also  important  to  know  what  this  error  is 
in  the  usual  case  of  the  two-span  bridge,  or  the  three-span  bridge  with  small  diagonals  or  no 
diagonals  in  the  centre  panel. 

After  the  sections  of  a  bridge  have  been  determined  from  calculations  based  on  the 
ordinary  formulae,  or  on  other  approximate  methods,  the  reactions  can  then  be  very 
accurately  determined  by  the  method  of  deflections  explained  below.  If  too  great  an  error 
is  found,  the  sections  can  be  corrected  accordingly.  This  would  change  the  reactions  slightly, 
but  not  enough  to  affect  seriously  the  stresses.  In  fact,  the  reactions  as  found  by  the 
formulae  will  usually  be  accurate  enough. 

We  will  consider  first  a  bridge  of  two  equal  spans  and  assume  the  bridge  fully  loaded. 
The  reactions  are  determined  as  follows:  Assume  first  the  bridge  loaded  with  the  given  load 
and  supported  only  at  the  centre,  both  arms  deflecting  equally.    Determine  the  deflection  of 

Pul 

the  end  point  by  the  formulaf  D='2^^,'\x\  which      =  deflection  of  the  end  point;  P  = 

stress  in  any  member  due  to  the  assumed  load  ;  u  =  stress  in  the  same  member  due  to  a  load 
of  one  pound  applied  at  the  end  ;  /  =  length  of  member;  E  —  modulus  of  elasticity ;  and  a  = 
area  of  cross-section  of  the  member. 

Now  find  in  the  same  way  the  upward  deflection  of  the  end  point  due  to  a  load  of  I  lb. 
applied  upwards  at  the  end.    The  stress  in  each  member  due  to  this  load,  and  which  corre- 

*  This  article  added  in  Sixth  Edition.    The  student  may  have  to  skip  the  discussion,  reading  only  the  conclusions 
on  p.  196(5,  until  after  Chapter  XV  has  been  read. 
\  See  p.  220  for  a  demonstration  of  this  formula. 


SWING  BRIDGES. 


spends  to  P  in  the  above  formula,  will  be  u,  the  value  of  which  is  already  known.  This 
upward  deflection  will  then  be     =  necessary  reaction  to  produce  an  upward  de- 

D 

flection  equal  to  D,  or  to  bring  the  end  back  to  normal  position,  will  be  7?  =  • 

Pul     ^  «V 

For  a  symmetrical  truss  symmetrically  loaded  it  is  necessary  to  sum  the  terms       and  ^ 

for  one  half  the  truss  only.  For  symmetrical  trusses  unsymmetrically  loaded  we  can  first 
assume  symmetrical  loads  and  determine  the  corresponding  reactions  ;  then  from  these  get  the 
reactions  for  the  unsymmetrical  loading  by  use  of  the  principle  that  the  moment  at  the  centre 
due  to  a  load  on  one  span  is  one  half  the  moment  due  to  two  such  loads  symmetrically  placed. 
Thus  let  M  =  centre  moment  for  full  load,  and  M'  =  centre  moment  for  one  span  only 
M 

loaded,  =  — .  If  R  and  R'  are  the  corresponding  reactions,  L  —  span  length,  and  2Fa  = 
Summation  of  moments  of  the  loads  on  one  span  about  the  centre,  we  have 

M=RL-  2Fa,    and    M'  =  R'L  -  2Fa ; 

M  R  2Pa 

and  since  M'  =  — ,  there  follows         R'  =  —  -\- 

2  Pa 

Now,  since         is  independent  of  any  errors  in  reactions,  the  actual  error  in  R'  must  be 

one  half  that  in  R.  That  is,  the  error  in  reaction  due  to  the  use  of  the  formula  is  twice  as 
great  for  bridge  fully  loaded  as  for  one  span  only  loaded.  As  these  two  cases  determine  the 
maximum  stresses  in  nearly  all  the  members,  it  will  not  usually  be  necessary  to  determine  true 
reactions  for  other  partial  loads.  However,  if  thought  desirable,  the  reactions  can  be  found  for 
each  panel  load  and  the  results  combined  to  form  influence  lines.  The  additional  labor  involved 
is  not  so  great  as  would  seem  to  be  the  case,  for  many  of  the  terms  involving  P  disappear. 

Pu/  uH 

For  a  bridge  of  unequal  spans  it  is  necessary  to  sum  the  terms  ■2'^^  and  -^"-^  for  all 

members  of  the  bridge.  In  this  case  it  is  convenient  to  assume  the  truss  supported  at  the 
centre  and  at  one  end  and  determine  the  deflections  and  reaction  at  the  other. 

The  foregoing  method  is  applicable  not  only  to  a  two-span  bridge  but  also  to  a  three- 
span  bridge  in  which  the  web  members  are  omitted  in  the  centre  span  or  panel ;  for,  assuniing 
such  a  truss  supported  at  the  centre,  or  at  the  centre  and  at  one  end,  the  stresses  due  to  any 
load  are  fully  determinate. 

To  get  some  notion  of  the  accuracy  of  the  ordinary  formulas,  reactions  have  been  com- 
puted for  several  cases  by  the  above  method  and  also  by  the  use  of  the  formulas ;  the  results 
are  given  below.  In  every  case  a  load  of  unity  per  lineal  foot  extending  over  the  entire 
bridge  has  been  assumed.    The  following  bridges  have  been  treated  : 

i^A)  The  Winona  Bridge,  a  description  of  which  will  be  found  in  Eiigineering  News,  Vol. 
XXVI,  1891,  p.  370.  This  bridge  is  similar  in  form  to  Fig.  229.  Each  span  consists  of  seven 
30-ft.  panels;  the  centre  panel  is  20  ft.  long,  centre  height  50  ft.,  and  end  height  25  ft. 

{B)  A  series  of  designs  with  the  half-truss  containing  6,  5,  4,  3,  and  2  panels  successively. 
The  general  dimensions  of  these  trusses  were  chosen  by  taking  a  corresponding  number  of 
panels  of  the  Winona  Bridge.  Cross-sections  of  members  were  properly  determined  from 
stresses  due  to  certain  assumed  dead  and  live  loads.  In  each  of  these  cases  computations 
were  made  for  trusses  with  and  without  a  central  panel. 

{C)  The  Milwaukee  drawbridge  of  the  C,  M.  &  St.  P.  Ry.,  which  is  described  in  the 
Engineering  and  Building  Record,  Vol.  XVI,  1887,  p.  747.  This  is  of  the  form  shown  in 
Fig.  230.  Each  span  has  five  18.5  ft. -panels ;  the  centre  panel  is  18  feet  long,  centre  height 
34  ft.,  and  height  at  second  panel-point  from  end  of  20.4  feet. 

The  results  of  these  computations  are  given  in  the  following  table ; 


MODERN  FRAMED  STRUCTURES. 


Truss. 

Reactions. 

True. 

By  Form. 

From 
Chords. 

(^i) 

(«a) 

64-3 

65-9 

64-3 

67.4 

52.9 

Is  "   

55-7 

57-7 

53-1 

56-3 

44.1 

45-8 

47.0 

42.0 

44-7 

34-7 

(^)U  "   

35-2 

36.1 

30.9 

33-1 

25.3 

|3  "   

24-3 

25.2 

20.0 

22. 1 

14.7 

12  "   

II. 8 

II. 9 

9.4 

II  .0 

8.5 

(O  

26.0 

29.0 

20.7 

Columns  headed  id)  are  the  reactions  for  the  several  trusses  with  the  central  panel 
omitted,  while  columns  headed  (U)  are  the  reactions  for  the  trusses  having  this  panel.  The 
column  headed  "  Reactions  from  Chords  "  contains  the  reactions  as  determined  by  deflec- 
tions, but  in  the  computation  of  which  the  chord  members  are  alone  considered.  These  have 
been  figured  only  for  trusses  without  centre  panel,  and  should  therefore  be  compared  with 
columns  {a).  These  reactions  are  the  same  as  would  be  found  by  the  use  of  fortnulas  which 
take  strict  account  of  the  variation  of  the  moment  of  inertia  of  the  truss. 

The  principal  points  brought  out  by  this  table  are : 

First,  the  comparatively  close  agreement  between  the  true  reactions  and  those  found  by  the 
ordinary  formulce. 

Second,  the  fact  that  the  fonmclce  taking  account  of  the  central  panel  give  results  ustially 
more  accurate  for  these  cases  than  the  formulce  which  neglect  this  panel  {compare  cobimns 
and  [b.)  with  (/;,)). 

Third,  that  the  reactions  found  by  treating  chords  only  are  in  every  case  much  more  ifi  error 
than  those  found  by  the  ordinary  formulce. 

No  attempt  has  been  made  to  determine  the  law  of  variation  of  the  error  in  the  ordinary 
formulae.  Such  a  generalization  would  be  very  difficult  to  make,  and  in  view  of  the  ease  with 
which  the  reactions  for  any  particular  case  can  be  checked,  it  would  be  unprofitable  as  well. 

Temperature  Effects. 

A  very  important  question,  and  one  of  especial  significance  when  discussing  the  accuracy 
of  working  formulae  is  that  of  the  effect  of  a  variation  of  temperature  between  different 
members  of  a  swing-bridge.  The  effect  on  reactions  of  any  given  difference  of  temperature 
can  very  quickly  be  found  from  our  previous  computations. 

Let  t  —  change  of  temperature  of  any  member ;  c  —  coefificient  of  expansion,  =  .0000065 
per  1°  F. ;  /  =  length  of  member.  Then  ctl  =  total  change  of  length  of  any  member.  The 
deflection  of  the  end  of  the  truss  due  to  the  change  of  length  of  this  member  will  be  uctl, 
where  u  is  the  same  as  in  our  previous  work,  and  for  a  change  of  temperature  of  any  number 
of  members  the  deflection  will  be 

Dt  2uctl. 

The  reaction  necessary  to  produce  a  like  deflection,  or  the  reaction  produced  by  the  assumed 
temperature  changes,  will  be 

^'=#.  w 

in  which  d  —  deflection  due  to  a  one-pound  load,  as  before.  For  example,  suppose  the  lower 
chord  of  tlie  Winona  Bridge  to  drop  in  temperature  1°  F.  In  this  case  ctl  for  each  member 
=  .00234  and  2ii  =  20,  whence 

Dt  =  .047,    and    Rt  —  — —  =  1400  lbs., 
'  .0000334 

a  very  large  amount  for  a  change  of  temperature  of  only  1°.  Considering  a  possible  difference 
of  temperature  of  30°  or  even  more,  the  great  effect  of  this  element  can  be  appreciated.  The 
effect  is  so  great  as  to  evidently  render  much  refinement  in  calculation  entirely  useless,  and  to 
indicate  that  the  ordinary  formulce  are  accurate  enough  for  all  ordinary  cases. 


CANTILEVER  BRIDGES, 


197 


CHAPTER  XIII. 
CANTILEVER  BRIDGES. 


179.  General  Considerations. — The  first  cantilever  bridge  built  in  America  was  the 
Kentucky  River  bridge,  by  C.  Shaler  Smith,  built  in  1876-7.    A  sketch  of  this  bridge  is  shown 


Fig.  232. 

in  Fig.  232.  The  bridge  is  continuous  from  E  to  F,  and  at  these  points  are  hung  the  ends  of 
two  simple  trusses  AE  and  ED. 

Since  this  bridge  was  constructed  several  very  large  cantilever  bridges  have  been  built 
in  America  and  elsewhere,  notably  the  great  Forth  bridge.  Fig.  233,  the  Niagara  bridge.  Fig. 
235,  the  Poughkeepsie,  Red  Rock,  Memphis,  and  others.    In  the  Niagara  bridge  the  suspended 


Fig.  234. 


span  DE  is  hung  from  the  ends  of  two  double  cantilevers,  AD  and  EH,&z.c\\  of  which  has  two 
points  of  support,  one  at  the  abutment  (not  shown  in  the  figure)  and  one  at  the  pier.  There 
being  no  diagonals  in  the  panels  over  the  piers,  the  trusses  are  free  to  turn  at  those  points  as  if 
supported  on  a  single  pin.  Fig.  234  is  a  form  of  bridge  proposed  by  C.  B.  Bender,*  as  being 
far  more  economical  than  the  Forth  bridge  as  built.  The  suspended  span  consists  of  a  three- 
hinged  arch  ACB;  the  arm  BD,  Fig.  233,  is  replaced  here  by  the  back-stay  EE  ior  anchor- 
age, and  the  roadway  is  supported  on  iron  trestle-work,  the  span  BD  being  a  land  span. 
Fig.  236  shows  another  form,  suitable,  however,  only  for  short  spans. 

The  chief  advantage  of  the  cantilever  bridge  over  a  single  span  bridge  is  in  being  able  to 
erect  the  overhanging  arms  and  the  su.spended  span  without  the  use  of  false  work.    Tn  the 


*  "  Principles  of  Economy  in  the  Design  of  Metallic  Bridges."     New  York,  John  Wiley  &  Sons. 


T98 


MODERN  FRAMED  STRUCTURES. 


Niagara  bridge,  for  example,  the  erection  is  carried  on  from  each  pier  to  the  centre,  at  least  three 
of  the  four  pieces,  DK,  LM,  EN',  and  OP,  connecting  the  suspended  span  to  the  cantilever, 
being  provided  with  adjustable  wedges  during  erection.  When  connection  is  made  at  the  centre, 
these  wedges  are  taken  out  and  the  central  span  swings  free  on  the  hangers  Z> J/ and  EO,  except 


Fig.  235. 

as  to  being  prevented  from  lorigitudinal  vibration  by  the  single  remaining  piece  or  a  similar 
device  holding  it  to  one  of  the  cantilevers.  Where  a  series  of  cantilever  spans  are  constructed, 
each  alternate  span  can  thus  be  erected  without  falsework.  Other  than  this,  the  cantilever 
possesses  no  advantage  over  discontinuous  spans  except  perhaps  for  very  long  spans. 


Fig.  236. 


For  economy  the  suspended  span  should  be  made  about  four-tenths  of  the  total  opening. 
However,  the  longer  this  span,  compared  to  the  total  span,  the  less  will  be  the  deflection,  which 
is  quite  an  important  consideration.  Thus  the  maximum  vertical  movement  of  the  hinge  E 
in  the  Niagara  bridge  under  the  test  load  was  about  9  in.,  and  of  the  point  E  in  the 
Kentucky  River  bridge  was  only  3^^  in.  The  Eighteenth  Street  viaduct  in  St.  Louis,  of  the 
form  of  Fig.  236,  vibrates  excessively  under  the  lightest  loads. 


Fig.  237. 

Fig.  237  is  a  proposed  form  of  bridge  for  a  long  series  of  equal  spans  where  erection  is 
difificult.    It  can  be  erected  entirely  without  falsework. 

Fig.  237  (a)  shows  a  bridge  proposed  for  the  English  Channel,  designed  by  Messrs. 


-187.57, 


-500  m- 


FlG.  237a. 


Schneider  &  Co.  and  H.  Hersent,  Sir  John  Fowler  and  Sir  Benjamin  Baker  consulting 
engineers. 

180.  Analysis. — Dead  Load  Stresses. — The  stresses  in  the  suspended  span  are  found  in 
the  same  way  as  for  any  single  span  truss.    In  both  forms  of  cantilevers,  EF,  Fig.  232,  and 


CANTILEVER  BRIDGES. 


199 


EH,  Fig.  235,  there  are  but  two  supports;  hence  the  reactions  due  to  any  load  are  readily 
found.  These  being  known,  the  moment  and  shear  at  any  section  can  be  found,  and  thence 
the  stresses.  Where  double  systems  of  web  members  are  used,  each  load  is  assumed  to  be 
carried  by  the  system  to  which  it  belongs.  At  points  of  connection  between  the  two  systems, 
the  loads  applied  to  the  truss  may  be  considered  as  equally  divided  between  the  syst'ems. 

Live  Load  Stresses. — Cantilever  bridges  being  used  mainly  for  long  spans,  double  systems 
of  web  members  are  quite  generally  adopted  for  the  sake  of  economy.  It  has  been  shown  in 
Chap.  V  that  with  double  systems  in  single  span  bridges  it  is  impracticable  to  use  the  exact 
wheel  load  method  in  computing  stresses,  and  that  some  conventional  method,  such  as  an 
equivalent  uniform  load,  or  a  uniform  load  with  one  or  two  excesses,  should  be  used  instead. 
It  is  still  more  difficult  to  apply  the  wheel  load  method  to  cantilever  bridges  with  double  sys- 
tems, and  hence  the  following  discussion  will  treat  mainly  of  uniform  loads,  with  or  without 
excess  loads. 

In  finding  the  proper  positions  of  live  load  for  maximum  stresses  in  the  various  members, 
it  will  be  useful  to  draw  the  influence  lines  for  moment  and  shear  in  a  single  intersection 
cantilever  of  the  type  shown  in  Fig.  232.    Fig.  238  shows  such  a  truss. 

1st.  Moment  at  G. — Fig.  {a). — A  load  unity  at  E  causes  a  negative  reaction  at  C  equal  to 


Fig.  238. 


^4  xl 

-J,  and  hence  a  negative  moment  at  G  equal  to       which  is  laid  off  as  E' e'  in  Fig.  {a).   As  the 

load  moves  to  D  or  F,  the  moment  at  G  decreases  uniformly  to  zero  and  the  influence  line  for 
this  portion  is  F'e'D'.    As  the  load  moves  from  D  to  G,  the  moment  increases  from  zero  to  a 

—  •*■) 

value  of   at  G ;  beyond  G  the  moment  decreases  again,  becoming  zero  when  the  load 

/, 

IS  at  C,  then  —  x)  when  at  B,  and  finally  zero  for  load  at  A.  Since  the  ratio  of  e'E'  to 
g'G'  is  equal  to  ^ZT^,  it  follows  that  g' D' e'  is  a  straight  line.    Similarly, g'C'b'  is  a  straight  line. 


\ 


200 


MODERN  FRAMED  STRUCTURES. 


The  influence  line  shows  that  for  a  maximum  positive  moment  at  G  the  span  CD  should 
alone  be  loaded,  and  that  for  a  maximum  negative  moment  the  spans  AC  and  DF  should  be 
loaded.  A  single  excess  in  the  two  cases  should  be  placed  at  and  dX  E  ox  B  according  as 
E'e'  or  B'b'  is  the  larger.  A  second  excess  a  fixed  distance  from  the  first  should  be  placed  in 
each  case  on  the  longer  segment  of  the  span  from  the  first  excess. 

2d.  Shear  in  the  panel  Gff.  —  Fig.  {b). — The  portion  D"h"g"C"  is  the  same  as  for  a  discontin- 
uous span.    Between  D  and  F  the  shear  in  GH  h  equal  to  the  reaction  at  (7  caused  by  the  load, 

and  is  negative,  having  a  value  of  j  for  unit  load  at  E.    When  the  load  is  between  A  and  C 

the  shear  is  positive  and  equal  to  the  negative  reaction  atZ>.  As  before,  the  lines  b"C"g"  and 
h" D"e"  are  straight  lines.  The  position  of  loads  for  a  maximum  positive  or  negative  shear  is 
evident  from  the  diagram.  If  full  joint  loads  only  are  considered,  then  for  maximum  positive 
shear,  for  example,  all  joints  from  H  io  D  in  span  CD,  and  span  AC,  should  be  fully  loaded. 

3d.  Moment  at  I.- — Fig.  {c). — For  loads  on  EE,  the  load  given  over  at  E,  and  hence  the 
negative  moment  at  /,  varies  directly  with  the  distance  of  the  load  from  F.  The  moment  at 
/  when  the  load  is  at  /,  is  zero.  Hence  the  influence  line  is  r"e"'F"',  in  which  E"'e"'  —  ly^x' 
The  condition  for  maximum  moment  is  the  same  as  for  moment  at  in  a  discontinuous 
truss  whose  span  is  equal  to  IF. 

4th.  Shear  in  the  panel  DI. — As  the  load  moves  from  F  to  E  the  positive  shear  increases 
uniformly  from  zero  to  a  value  equal  to  unity  with  unit  load  at  E.  The  shear  then  remains 
constant  until  the  load  passes  /,  then  decreases  to  zero  as  the  load  reaches  the  next  panel 
point  to  the  left  (not  the  abutment,  necessarily).    The  position  for  a  maximum  is  evident. 

The  foregoing  influence  lines  show  in  a  general  way  the  positions  of  loads,  whether  uni- 
form, or  uniform  with  one  or  two  excesses.  If  exact  methods  are  desired  for  a  single  inter- 
section truss,  the  conditions  for  maxima  can  be  readily  written  out  from  the  influence  lines 
Thus  for  maximum  positive  moment  at  G,  if  G^,  G^,  G^,  and  G^  are  the  sums  of  the  loads 
on  AB,  BG,  GE,  and        respectively,  we  have  at  once  the  condition  that 

G^  tan  a-,  -j-  G^  tan      —  G^  tan      —  G^  tan 

must  pass  through  zero  by  passing  from  positive  to  negative  as  the  loads  are  moved  to  the 
left.    The  values  of  the  tangents  of  the  angles  are  readily  substituted  in  any  case. 

The  influence  lines  for  moments  and  shears  in  the  shore  arm  GH,  Fig.  235,  are  the  same  as 
those  for  the  span  CD  above  considered,  except  that  the  portion  to  the  right  of  Z>  does  not  exist. 

The  position  of  the  load  having  been  found  in  any  case,  the  reactions  and  stresses  are 
found  as  for  dead  load. 

181.  Example.  Indiana  and  Kentucky  Bridge. — Fig.  239. — The  span  lengths  CD  and 
DE  are  given  only  approximately. 


Fig.  239. 

Dead  Load  Stresses. — The  stresses  in  the  span  CD  due  to  loads  between  C  and  D  are 
found  as  in  Art.  78,  Chap.  IV. 

To  find  the  stresses  in  CD  due  to  loads  on  AC  and  DF,  first  compute  one  of  the  abut- 
ment reactions  at  C  or  D,  and  the  moment  at  this  abutment.  Then  the  shear  just  to  the  right 
of  C,  for  example,  is  equal  to  the  reaction  at  C,  minus  the  loads  on  BC.  This  shear  is  constant 
from  C  to  D,  the  loads  on  CD  not  being  considered,  and  may  be  assumed  to  be  equally 


CANTILEVER  BRIDGES. 


20I 


divided  between  the  two  web  members  cut  by  any  section.  The  web  stresses  due  to  exterior 
loads  are  thus  all  equal.  If  AC  is  symmetrical  to  DF,  then  these  web  stresses  are  zero.  Since 
the  horizontal  components  of  the  stresses  in  KN  and  LM,  for  example,  are  equal  and  opposite 
in  direction,  the  stresses  in  KL  and  MN  are  equal  and  may  be  found  by  taking  moments 
about  o  of  the  forces  between  section  and  the  section  p'q'  just  to  the  right  of  C.  The 
only  forces  acting  on  the  left  of  J>g,  besides  the  stresses  in  KL  and  MN,  are  the  moment  and 
shear  at  />'g'  which  are  already  known.  This  moment  being  due  to  a  couple,  its  value  at  o  is 
the  same  as  at  pq,  and  adding  the  moment  of  the  shear  about  o  we  have  the  total  moment  to 
be  resisted  by  KL  and  MN. 

The  stresses  in  the  arm  BE  due  to  loads  on  that  arm,  other  than  the  loads  at  P  and  R, 
are  found  as  in  Art.  78,  Chap.  IV;  that  is,  by  considering  the  web  systems  independent  and 
all  the  load  applied  at  the  main  panel  points,  [/,  V,  etc.,  the  loads  at  W,  X,  etc.,  being  con- 
sidered as  carried  to  the  points  U,  V,  etc.,  by  separate  small  trusses  UwV,  etc.  The  stresses 
in  these  small  trusses  are  added  to  the  stresses  in  those  main  members  which  coincide  with 
the  members  of  the  small  trusses. 

The  stresses  in  P(>aiid  PR  are  found  from  the  load  at  P,  which  is  one-half  the  weight 
of  the  truss  EF.  The  vertical  component  of  the  stress  in  PR  together  with  the  load  at  R 
may  be  assumed  to  be  divided  equally  between  the  two  web  members  ZR  and  QR.  The 
stresses  in  the  remaining  members  are  then  found  by  separating  the  systems  and  proceeding 
in  the  ordinary  manner. 

Live  Load  Stresses. — For  the  maximum  positive  moments  in  span  CD  this  span  should  be 
fully  loaded.  If  a  uniform  load  is  used,  the  stresses  are  found  in  the  same  way  as  for  dead 
load.  If  an  excess  load  is  used,  the  position  of  such  load  for  maximum  stress  is  to  be  found. 
For  piece  KL,  the  centre  of  moments  for  one  web  system  is  at  M,  and  for  the  other  is  at  N. 
The  excess  load  should  be  at  that  one  of  these  points  nearest  the  centre  of  the  span,  the  sub- 
vertical  oO  not  being  considered  in  finding  upper  chord  stresses.  If  a  second  excess  is 
employed,  it  should  be  on  the  longer  segment  of  the  span  from  the  first. 

The  position  of  loads  being  known,  the  stress  in  KL  is  found  by  assuming  independent 
systems  and  neglecting  the  sub-verticals,  or  treating  them  as  parts  of  trussed  beams,  simply 
transferring  the  intermediate  panel  loads  to  the  main  panel  points.  For  the  member  MN, 
any  single  excess  should  be  placed  at  O,  for  the  tension  in  MN  is  due  both  to  the  main  truss 
and  to  the  truss  MoN.  A  second  excess  should  be  placed  on  the  longer  segment  of  the  span 
from  O.  The  stress  in  J/tV  due  to  the  main  truss  is  found  by  considering  the  systems  inde- 
pendent, one-half  the  load  at  O  going  to  M  and  one-half  to  N.  To  this  stress  is  added  the 
stress  from  the  truss  MoN,  with  excess  at  O. 

For  the  maximum  negative  moments  in  CD,  the  spans  AC  and  DF  dive.  loaded,  with  one 
excess  at  B  or  E,  and  the  other  (if  used)  outside  or  inside  these  points  according  as  the 
suspended  spans  are  longer  or  shorter  than  the  cantilevers.  The  chord  stresses  in  CD  arc 
found  as  for  dead  load. 

For  the  maximum  tension  in  Ko  and  compression  in  Ko' ,  the  span  AC  should  be  fully 
loaded  and  CD  loaded  from  O  to  D.  One-half  the  load  at  O  goes  to  A'',  and  one-half  to  M 
and  into  the  other  system.  Considering  all  loads  to  the  right  of  N  as  applied  at  the  main 
panel  points,  the  vertical  components  in  Ko  and  Ko'  due  to  loads  on  CD  are  equal  to  the  shear 
in  panel  N'N  o{  the  system  to  which  these  members  belong.  The  stresses  in  Ko  and  Ko'  due 
to  loads  on  AC  is  found  as  for  exterior  dead  load.  Any  excess  should  be  placed  either  at  B 
or  N,  whichever  vi^ill  give  the  greater  stress  as  may  be  found  by  trial.  It  will  usually  be  at  N, 
since  if  at  B  its  effect  is  divided  between  Ko  and  oM.  For  the  piece  c;A''the  interior  loading 
should  extend  from  N  to  D,  since  a  load  at  O  will  cause  a  greater  compression  in  oA^  due  to 
truss  MoN  than  tension  due  to  the  additional  one-half  panel  load  at  N.  The  excess  if  on  CD 
should  be  at  N.    For  maximum  compression  in  o'N',  panel  points  O'  to  D  inclusive  should 


202 


MODERN  FRAMED  STRUCTURES. 


be  loaded,  for  the  load  at  O'  causes  a  compression  in  o' N'  as  part  of  the  truss  iVV^  which  is 
greater  than  the  tension  in  o'N'  due  to  the  additional  half  load  at  N'.  The  excess  should  be 
at  N.  The  stresses  in  ^^A^and  o' N'  due  to  loads  on  AC  are  the  same  as  that  in  Ko.  Stresses 
due  to  negative  shear  in  the  span  CD  are  found  in  a  similar  way,  the  span  DF  and  the  portion 
of  CD  to  the  left  of  the  members  in  question  being  loaded. 

For  maximum  negative  moments  in  the  arm  D£,  the  span  DF  may  be  considered  fully 
loaded,  since  the  loads  to  the  left  of  the  centre  of  moments  have  no  effect.  Any  single  ex- 
cess should  be  placed  at  E ;  and  if  a  second  excess  is  employed,  it  should  be  placed  to  the 
right  or  left  according  as  EF  or  ED  is  the  longer.    The  stresses  are  found  as  for  dead  load. 

The  web  stresses  in  DE  are  also  found  as  for  dead  load.  The  excess,  if  any,  should  be 
placed  as  for  positive  shear  in  CD. 

182.  Wind  Stresses. — Wind  pressure  is  carried  to  the  abutments  by  means  of  horizontal 
lateral  bracing  arranged  in  the  same  way  as  the  main  vertical  trusses.  The  wind  stresses  are 
then  found  in  a  way  precisely  similar  to  that  explained  in  the  preceding  articles. 


ARCH  BRIDGES.  203 


CHAPTER  XIV. 
ARCH  BRIDGES. 


GENERAL  PRINCIPLES. 

183.  Arches  of  metal  may  consist  of  curved  beams  with  solid  webs  and  flanges,  or  they 
may  be  curved  trusses  with  upper  and  lower  chords  and  web  members,  either  riveted  or  pin- 
connected. 

With  reference  to  the  ordinary  modes  of  support  arches  may  be — 

1st.  Hinged  at  the  abutments  and  at  the  crown. 

2d.  Hinged  at  the  abutments  and  continuous  throughout. 

3d.  Fixed  rigidly  to  the  abutments  and  continuous  throughout. 

In  the  first  case  the  arch  consists  of  two  separate  framed  structures,  and  the  reactions 
and  stresses  can  be  found  by  ordinary  methods  of  statics.  In  the  other  cases,  however,  the 
reactions  depend  not  only  upon  the  loads  but  also  upon  the  form  and  material  of  the  arch. 
In  finding  these  reactions  the  arch  will  in  all  cases  be  treated  as  a  simple  curved  beam  with  a 
constant  or  variable  moment  of  inertia  as  the  case  may  be.  (By  a  curved  beam  is  meant  one 
which  has  a  curved  form  in  its  natural  or  unstrained  condition.)  All  loads  will  be  treated  as 
vertical.  The  general  method  of  treatment  is  the  same  as 
that  employed  by  Prof.  Greene  in  his  "  Trusses  and  Arches," 
Part  III, and  by  Prof.  DuBois  in  his  "  Framed  Structures," 

184.  Relation  between  the  Equilibrium  Polygon 
and  the  Stresses  at  any  Section  of  an  Arch. — Let  AB 
Fig.  240,  be  an  arch  of  two  hinges,  with  loads  P^,  P^,  and  a" 
P^.    Each  abutment  reaction  will  have  a  horizontal  com- 
ponent and  a  vertical  component.    From  ^  hor,  comp. 

=  o  we  know  that  these  horizontal  components  are  equal.  o< 
Suppose   that   in   some  way  these  abutment  reactions 
have  been  found,  the  force  diagram,  01234560, 
drawn,  and  also  the  corresponding  equilibrium  polygon 
AbcdB.    Notice  that  here  in  the  case  of  an  arch,  the  equal 


Fig.  240. 


and  opposite  forces  //,  and  //,  are  not  imaginary,  as  was  the  case  for  the  rigid  frames  treated 
of  in  Chapter  II. 

According  to  the  principles  of  Chap.  II,  Art.  28,  any  segment,  be,  of  the  equilibrium  poly- 
gon is  the  line  of  action  of  the  resultant  of  all  the  forces,  H^,  F, ,  and  P, ,  to  the  left  (or  right), 
this  resultant  being  given  in  amount  and  direction  by  the  ray,  0-3,  in  the  force  polygon  to 
which  the  segment  is  parallel.  The  stresses  at  any  section  of  the  arch,  therefore,  between  the 
loads  and  P^  are  the  same  as  would  result  if  77, ,  F, ,  and  were  replaced  by  a  single  force 
0-3  applied  in  the  line  be.  Fig.  241  represents  a  portion  of 
the  arch  of  Fig.  240,  to  the  left  of  such  a  section.  The  force 
0-3,  =  R,  applied  in  the  line  be,  is  in  equilibrium  with  the 
stresses  at  the  section.  These  stresses  may  be  considered 
as  consisting  of  a  uniformly  distributed  direct  stress  or  thrust 


h  ^0-3- 

~F  

si" 


a 


Fig.  241. 


MODERN  FRAMED  STRUCTURES. 


T,  in  the  direction  of  the  tangent  at  N,  a  shear  S,  at  right  angles  to  T,  and  a  bending 
moment  M.    If  a  is  the  angle  between  R  and  the  tangent  at  N,  we  have 


R  cos  a  =z  T 


and 


R  sm  a  =  S. 


(I) 


(2) 


Also,  taking  centre  of  moments  at  N,  the  centre  of  gravity  of  the  cross-section,  we  have,  from 
Figs.  240  and  241, 

Hz  =  M,  (3) 


where  H  is  the  pole  distance  and  s  is  the  vertical  intercept  from  the  equilibrium  polygon  to 
the  centre  of  moments.  Thus  the  thrust,  shear,  and  moment  at  any  section  are  readily  found 
after  the  equilibrium  polygon  is  once  drawn. 

In  a  solid  beam  the  fibre  stresses  result  directly  from  the  above  equations.  In  a  braced 
arch,  however.  Fig.  242,  the  stresses  in  the  members  cut  by  any  section  are  more  readily  found 
by  the  ordinary  methods.  Thus  the  stress  in  AB  is  equal  to  the  moment  of  R  about  C, 
divided  by  the  lever  arm  o{  AB,  —  Hz'  -f-  d.  Likewise  the  stress  in  DC  —  Hz"  d.  The 
component  of  the  stress  in  AC  normal  to  the  arch  is  equal  to  the  shear,  =  R  sin  a.    If  the 


R 


Fig.  242.  Fig.  243. 


chords  AB  and  Z>^7are  not  parallel,  then  the  stress  in  AC  \s  found  by  getting  the  moment  of 
R  about  the  intersection  of  AB  and  DC,  and  dividing  by  the  lever  arm  of  AC.  In  a  built 
beam,  Fig.  243,  in  which  the  direct  stress  and  moment  are  taken  entirely  by  the  flanges 
and  the  shear  by  the  web,  the  stress  in  flange  A  is  equal  to  Hz'  d,  and  in  B  is  equal  to 
Hz"  -^  d.    The  shear  is  as  before  equal  to  R  sin  a. 

In  Chap.  II  it  has  been  shown  that  for  a  given  system  of  loads  an  infinite  number  of 
equilibrium  polygons  can  be  drawn  by  assuming  various  amounts  and  directions  for  one  of 
the  reactions;  that  is,  by  assuming  various  poles.  But  by  Art.  42,  Chap.  II,  three  points 
determine  an  equilibrium  polygon  ;  hence  if  we  have  given,  besides  the  loads,  three  points 
through  which  the  polygon  must  pass,  or  three  equations  of  condition  which  will  determine 
three  points,  the  equilibrium  polygon  is  at  once  determined,  and  the  true  reactions  and  stresses 
may  thence  be  found.  The  three  points  required,  or  the  necessary  condition  equations,  for 
the  three  kinds  of  arches  will  be  deduced  in  what  follows.  It  will  be  sufficient  for  our  pur- 
poses to  consider  but  a  single  load,  that  is,  our  polygon  will  consist  of  but  two  segments. 


ARCH  BRIDGES. 


205 


ARCH  OF  THREE  HINGES. 


185.  The  Equilibrium  Polygon.— Fig.  244. — In  this  case  we  know  that  since  the  arch 
is  free  to  turn  at  A,  B,  and  C,  the  moments  at  these 
points  must  be  zero ;  hence  the  polygon  passes  through 
these  points,  and  to  draw  it  we  need  only  produce  BC  to 
the  load  vertical  at  k  and  join  Ak. 

186.  Position  of  Loads  for  a  Maximum  Stress  in 
any  Member. — Let  pq,  Fig.  245,  be  any  section  cutting 
three  members  of  a  braced  arch  of  three  hinges,  A,  B, 
and  C.  The  centre  of  moments  for  DE  is  G,  and  a  load 
at  k,  the  intersection  of  BC  and  AG,  will  cause  no  stress 
jn  DE,  since  the  line  AG  \s  the  line  of  action  of  the  abut- 
ment reaction  at  A,  the  only  external  force  on  the  left  of 
the  section.  For  loads  between  k  and  B  the  reaction  line 
Ak  will  lie  below  G,  passing  through  C  for  loads  on  CB. 
The  moment  of  the  reaction  at  A,  about  G,  will  then  be 
negative  and  cause  tension  in  DE.  For  loads  between  k  and  G,  inclusive,  the  reaction  line 
Ak  lies  above  G,  and  there  is  then  compression  in  DE.   For  loads  between  D  and  A  the  only 


A 

(a)  J 

/  H 

% 

Fig.  244. 


A  - 


Fig.  245. 


force  on  the  right  of  the  section  is  the  reaction  at  B  which  acts  in  the  line  BC,  thus  causing 
compression  in  DE.  Therefore  for  a  maximum  tension  in  DE  the  load  should  extend  from 
k  to  B,  and  for  maximum  compression  it  should  extend  from  A  to  k.  The  loads  are  usually 
taken  as  uniformly  distributed,  and  are  applied  at  joints,  or  at  intervals  along  the  upper  flange 
in  the  case  of  the  flanged  beam. 

The  centre  of  moments  for  EG  is  at  D,  and  from  considerations  similar  to  the  preceding, 
it  is  found  that  the  maximum  tension  and  the  maximum  compression  in  this  member  occur 
when  the  load  extends  to  the  left  and  right,  respectively,  of  k' . 

For  the  web  member  DG  draw  Ak"  parallel  to  DE.  and  EG  if  these  members  are  parallel, 
or  to  their  intersection  if  not  parallel.  For  all  loads  between  G  and  B  the  reaction  line  at  A 
lies  above  Ak" ,  since  this  reaction  line  never  passes  below  C.  Hence  the  component  of  the 
reaction  perpendicular  to  Ak"  in  case  of  parallel  chords,  or  the  moment  of  the  reaction  about 
the  intersection  of  DE,  EG,  and  Ak"  in  case  of  non-parallel  chords,  produces  tension  in  the 
member  DG.  For  loads  from  D  to  A  the  right  reaction  acting  in  the  line  causes  compres- 
sion in  DG.  Hence  for  maximum  tension  in  DG,  GB  should  be  loaded,  and  for  maximum 
compression  DA  should  be  loaded.  If  Ak"  should  pass  to  the  left  of  C,  then  all  loads  between 
its  intersection  with  BC,  and  B,  would  cause  compression  in  DG. 

In  the  case  of  a  built  beam  the  centres  of  moments  are  taken  as  in  Art.  184,  and  in  find- 
ing the  loading  for  maximum  shear,  the  line  corresponding  to  Ak"  is  drawn  parallel  to  the 
flanges  at  the  section  considered. 


206 


MODERN  FRAMED  STRUCTURES. 


187.  Computation  of  Stresses. — The  position  of  the  loading  producing  the  maximum 

stress  in  any  member  having  been  found  according 
to  the  previous  article,  the  equilibrium  polygon 
may  be  constructed  for  the  entire  system  of  loads 
according  to  the  method  of  Art.  42,  Chap.  II,  and 
the  stress  in  the  member  found  as  in  Art.  184.  Or 
the  reactions  and  stresses  may  be  found  analytically. 
If  c  is  the  half-span,  Fig.  246,  r  the  rise,  and  b  the 
distance  of  any  load  P  from  the  centre  C,  measured  positively  towards  the  right,  we  have 


V,  =  P- 


2C 


and     V,  =  P-  V,=P 


f-\-b 


2C 


(4) 


By  taking  centre  of  moments  at  C  and  treating  the  structure  AC,  we  have  for  loads  on  CB, 

Hr=V,c, 


whence 


c       c  —  b 
H=  V,-  =  P-  . 


Similarly,  for  loads  on  .^C  we  have 


c        c  A-  b 
H=  V,-  =P^. 


(5) 


(5«) 


The  components  of  the  reactions  can  thus  be  computed  for  each  joint  load  and  the  results 
added.  The  reactions  being  known,  the  stresses  can  be  found  by  the  ordinary  method  of 
moments. 


CURVED  BEAMS. 

188.  Deflection  of  a  Curved  Beam.— Before  proceeding  further  it  will  be  necessary  to 
investigate  the  general  case  of  the  deflection  of  a  beam,  the  beam  to  be  of  any  shape,  and 
acted  upon  by  forces  all  lying  in  the  same  plane.  The  follow  ing  discussion  is  taken  mainly 
from  Prof.  Church's  "  Mechanics  of  Engineering,"  pp.  444-9. 

Let  AB,  Fig.  247,  be  any  portion  of  such  a  beam  in  its  unstrained  form.    Suppose  now 

that  under  the  action  of  certain  forces  the  beam  is 
brought  into  the  position  A'B',  the  change  in  position 
being  due  to  any  cause  whatever.  We  wish  now  to 
find  the  movement  of  A  and  the  tangent  to  the  neu- 
tral axis  at  A,  with  reference  to  B  and  the  tangent  at 
B,  these  two  points,  and  B,  being  any  two  points  in 
the  beam.  This  relative  motion  will  be  made  appa- 
rent by  making  B'  coincide  with  B,  and  the  tangent  at 
B'  coincide  with  the  tangent  at  B.  This  new  position 
is  represented  by  the  dotted  outline  A" B.  The  ab- 
solute movement  now  shown  is  the  relative  movement 
Fig.  247.  required.    The  tangent  at  A  has  moved  through  an 

angle  A^),  making  now  an  angle  with  the  tangent  at  ^  of  6* -(-  B  being  the  original  angle; 
the  point  A  has  also  moved  in  space  a  distance  AA" ,  the  components  of  which  motion,  referred 
to  any  two  rectangular  axes  with  origin  at  A,  will  be  called  Ay  and  Ax. 


ARCH  BRIDGES. 


207 


{d)  Change  of  Inclination  of  Tangent,  =  J0. — Let  CEFD,  Fig.  248,  be  an  element 
of  the  beam  of  length  ds,  whose  end  faces  CE  and  DF  are 
at  right  angles  to  the  axis,  and  whose  end  tangents  make  an 
angle  with  each  other  originally  equal  to  dd.  Let  d<l>  be  the 
change  in  angle  between  end  faces  or  end  tangents  due  to  bend- 
ing. The  change  in  length  of  a  fibre  at  a  distance/ from  the 
neutral  axis  will  be  equal  to  yd<i>,  and  the  corresponding  stress  per 

unit  area  will  be  equal  to/=  ,  where  E  is  the  modulus  of 

elasticity  of  the  material,  and  ds  is  the  original  length  of  the  fibre. 
If  da  is  an  element  of  area  of  the  cross-section,  the  total  moment 

of  resistance  of  the  beam  is  equal  to      fday  =  J*  Ey'da^ . 

But  for  any  particular  section,  E  and       are  constant ;  and  if  M 

IS  the  moment  of  the  external  forces  on  one  side  of  the  section  about  N,  and  /  is  the  moment 
of  inertia  of  the  section,  we  have 


Fig.  248. 


d<t> 


from  which  we  have 


,^  Mds 


and  in  Fig.  247 


(6) 


(7) 


{b)  Components  of  A's  motion,  =  Ay  and  Ax. — Let  AEDCB,  Fig.  249,  represent 
the  axis  of  the  unstrained  form  of  the  beam,  and  A''E"D'CB  the  strained  form  A"B, 
Fig.  247.    Now  conceive  the  beam  to  pass  into  its  strained  form  by  the  successive  bend- 


FiG.  249. 


ing  of  each  ds  in  turn.  The  bending  of  the  element  BC  through  the  angle  d<p,  causes 
the  portion  AC  to  turn  through  the  same  angle  d<p  about  C  as  a  centre,  with  radius  u,  the 
point     moving  to  A'  through  a  distance  dv,  having  the  components  dy  and  dx.    Then  from 


208 


MODERN  FRAMED  STRUCTURES. 


the  bending  of  DC  the  point  A'  moves  to  A'\  etc. 
point  C,  we  have,  by  sinmilar  triangles, 


If  X  and  y  are  the  co-ordinates  of  any 


dy  _x 
dv  II 


.     dx  y 

and    —  =  -. 
av  u 


Solving  for  dy  and  dx  and  substituting  for  dv  the  value  udcp,  we  have 

dy  -   xd(p    and    dx  =  yd(p. 
Substituting  the  value  of  d(f>  from  eq.  (6),  we  have 

Ay  =fdy         xd<p  -  f 


Mxds 


and 


Ax  =  J* dx  —  J*  ydcf)  =  J* 


Myds 
EI  ' 


(8) 


(9) 


Fig,  250. 


{c)  Application  of  eqs.  (7),  (8),  and  (9). — To  make  clear  the  application  of  the  above  equa- 
tions, let  us  take  the  case  of  a  two-hinged  arch  ACB,  Fig.  250,  supporting  the  loads  P^,  P^, 

etc.  The  full  line  ACB  represents  the  posi- 
tion of  the  arch  when  under  no  load,  and  the 
dotted  line  ACB  represents  its  form  when 
loaded.  The  tangents  to  the  curve  of  the 
arch  at  B,  in  the  two  positions,  are  BT  and 
BT'  respectively.  Now  the  total  movement 
of  any  point  in  the  arch  with  reference  to  B, 

S  ^r::-:_-::>bB  due  to  the  application  of  the  given  loads, 

may  be  divided  into  two  parts ;  one  a  move- 
ment about  ^  as  a  centre  due  to  the  turning 
of  the  arch  on  the  hinge  B,  and  the  other  a 
movement  due  to  the  bending  of  the  arch  between  the  given  point  and  B.  These  motions 
occur  simultaneously ;  but  if  we  conceive  them  to  take  place  successively,  the  first  motion  will 
bring  the  arch  into  the  position  A'C'B,  the  tangent  ^7"  moving  to  BT'  through  an  angle  ^, 
and  the  entire  arch  turning  about  ^  as  a  centre  through  the  same  angle.  Now  conceive  the 
bending  to  take  place.  The  arch  will  then  come  into  the  position  shown  by  the  dotted  line 
ACB, and  the  points' will  move  to  ^,  making  the  total  movement  of  A  equal  to  zero.  The 
movement  of  any  point  and  the  tangent  at  that  point  with  reference  to  B  and  BT',  due  to 
bending,  as  for  example,  the  point  A'  and  the  tangent  at  A',  is  now  given  by  eqs.  (7),  (8),  and 
(9).  Since  the  point  A'  comes  back  to  A,  its  motion  due  to  bending  is  equivalent  to  the  arc 
A' A,  this  arc  being  the  path  of  the  first  part  of  its  motion.  Hence  with  the  axis  of  X  taken 
parallel  to  AB,  the  distance  Ax  is  very  small  compared  to  Ay  (if  Ay  were  a  differential  of  the 
first  order.  Ax  would  be  of  the  second  order  and  therefore  zero  compared  to  Ay)\  and  since  Ay 
is  small  compared  to  the  dimensions  of  the  arch,  we  may  put  Ax  equal  to  zero.  Hence  from 
eq.  (9)  we  have  for  the  point  A 


Ax=/ 


Myds  _ 


(10) 


This  is  the  only  one  of  the  equations  (7),  (8),  and  (9)  whose  value  is  known  beforehand,  and 
hence  the  only  one  which  is  of  service  in  finding  reactions. 

In  the  case  of  an  arch  with  fixed  ends,  the  end  tangents  are  fixed  as  well  as  the  points  A 
and  B ;  hence,  for  each  end  referred  to  the  other,  each  of  the  equations  (7),  (8),  and  (9)  reduce 
to  zero. 


ARCH  BRIDGES. 


PARABOLIC  ARCH  OF  TWO  HINGES;    VARIABLE  MOMENT  OF  INERTIA. 

189.  The  Equilibrium  Polygon  * — In  Fig.  251,  let  r  =  rise ;  c  =  half-span  ;  b  =  distance 
of  any  load  from  the  centre,  measured  positively  towards  the  right ;  and  x  and  j/  =  the  co- 
ordinates of  any  point  of  the  arch  referred  to  A  as  the  origin.    Since  the  moments  at  A  and 


Fig.  251. 

B  are  zero  we  know  that  the  equilibrium  polygon  for  the  load  P  must  pass  through  these 

points.    We  also  have  the  further  condition  from  eq.  (10)  that   /  =  O. 

The  modulus  of  elasticity,  E,  will  be  taken  as  constant ;  and  if  H  is  the  pole  distance  and 
is  the  vertical  ordinate  between  the  equilibrium  polygon  and  any  point  E,  then  M 
and  hence  by  substitution 

r^Myds     H  r^zyds  zyds  ,  , 

If  now  we  make  the  further  assumption  that  /  increases  from  the  crown  to  the  springing 
line  in  the  same  ratio  as  sec  /,  where  i  is  the  inclination  of  the  arch  at  any  point  to  the  hori- 
zontal, we  have 

— =  a  constant  =  /, , 
sec  t 

where  /„  is  the  moment  of  inertia  at  the  crown.    This  assumption  is  a  reasonable  onef  and 
sufificiently  exact  for  practical  purposes. 
From  Fig.  251  we  have 

dx  =  ; 
sec  t 

hence,  by  substituting  the  above  value  of  /  in  eq.  (11),  we  have 

^"zyds  zyds  ^ 


n^zyds  zyds        ^   P  j 

J  A    I    ~  Ja  h  sec  ' 


whence 


000 


J*  zydx  —  o,    or         zydx  =  0.  ...... 

From  Fig.  251  we  have  z  =  y  —  z'  \  hence  eq.  (12)  becomes 

/  y'dx  —  /  z'ydx  =  o.   .....    ^   ..  o 

For  a  parabola,  with  origin  at      we  have 

r  —  y       r  rx 
J^Zr^.  =  -.,    whence   y  =  -  {2c  -  x).    .    .  . 

*  The  following  demonstration  and  that  of  Art.  194  are  from  Prof.  Greene's  "Trusses  and  Arches,"  Part  III,  pp 
44.  45.  60-63. 

f  Since  the  direct  stress  in  the  arch  for  a  full  load,  and  also  its  cross-section,  do  increase  from  crown  to  springing 
in  accordance  with  this  law. 


(12) 

(13) 
(14) 


210 


MODERN  FRAMED  STRUCTURES. 


Substituting  this  value  oi  y  in  the  first  term  of  eq.  (13)  and  integrating,  we  have 

fdx  =  {ac'x'  -  4cx'  +  x*)dx 


'      /  15 


^■M  '5 
For  the  second  term  of  eq.  (13)  we  have,  from  A  to  G, 


r'c. 


whence   2'  = 


(15) 


(16^ 


Substituting  from  (16)  and  (14)  in  (13)  and  integrating  from  o  to  (c-{-  6),  we  have 

=^.[2,(.+,,_(i±i):]  

To  integrate  the  portion  on  the  right  of  G,  we  may  take  our  origin  at  B.  The  integral 
will  be  then  the  same  as  eq.  (17),  but  with  {c  —  b)  put  for  {c  ~\-  b).    That  is, 

=  ^{i,.-.).-*!^]  (.8) 

Substituting  from  (15),  (17),  and  (18),  in  (13)  and  reducing,  we  have 


Solving  for     we  have 


16  ,      ryj  b\ 


This  equation  gives  the  value  of  in  terms  of  known  quantities  and  hence  determines 
the  third  point  in  the  equilibrium  polygon.  As  the  position  of  the  load  varies,  eq.  (19)  is  the 
equation  of  the  locus  of  the  point  k.  Hence  if  this  curve  be  constructed,  the  equilibrium 
polygon  for  any  load  is  drawn  by  simply  joining  the  points  A  and  B  with  the  intersection  of 
the  load-vertical  with  this  curve. 

190.  Position  of  Loads  for  a  Maximum  Stress  in  any  Member. — In  Fig.  252  the  locus 
of  the  point  k  is  the  curve  MN,  having  an  ordinate,^,,  equal  to       at  the  centre,  and  ffr  at 


A  B 

Fig.  252. 

the  ends  of  the  arch.  For  the  piece  CD,  with  centre  of  moments  at  F,  a  load  at  k  produces 
no  stress,  while  all  loads  to  the  right  produce  tension  and  all  loads  to  the  left  compression. 
For  FE,  all  loads  to  the  right  of  k'  cause  compression  and  all  loads  to  the  left  tension.  For 


ARCH  BRIDGES. 


211 


the  web  member  DF,  draw  Ak"  towards  the  intersection  of  CD  and  FE.  Then  loads  between 
D  and  k"  cause  positive  shear  on  section  pq,  or  compression  in  DF,  while  loads  to  the  right  of 
k"  and  to  the  left  of  F  cause  tension  in  DF.  For  such  a  piece  as  EI,  with  centre  of  moments 
at  G,  loads  between  k'"  and  produce  positive  moment  or  tension  in  EI,  while  loads  on  the 
remaining  portions  produce  compression. 

The  loading  for  the  maximum  stress  in  any  member  is  thus  readily  found,  after  having 

y 

constructed  the  curve  MN.  The  following  table  gives  enough  values  of  to  enable  this 
curve  to  be  constructed  with  sufificient  accuracy  for  this  purpose. 

TABLE  OF  VALUES  OF-"  FOR  VARIOUS  VALUES  OF-. 

r  c 


b 

0.8 

c 

o.o 

0.2 

0.4 

0.6 

1 .0 

1.280 

1.290 

1.322 

1.379 

1.468 

1.600 

r 

191.  Computation  of  Stresses. — The  stress  in  each  member  due  to  each  load  maybe 
found  graphically  by  computing  and  drawing  the  equilibrium  polygon  for  each  load,  the 
stresses  resulting  as  explained  in  Art.  184.  In  this  case,  however,  it  will  be  as  easy  to  deter- 
mine the  stresses  analytically. 

In  Fig.  251  we  have,  by  taking  moments  about  B  and  then  about  A, 


V,  =  P'—    and     V,  =  P'-^.  (20) 


Also,  by  similar  triangles  in  Figs.  251  and  251 


H  _c  +  b 

whence,  from  eq.  (20), 


H=v^±±=pt^.  .  . .  :  (a,) 

J.  2cy, 


By  means  of  eqs.  (19),  (20),  and  (21)  the  components  of  the  reactions  due  to  each  load 
can  be  computed  and  the  results  added  for  those  loads  which  act  together  to  cause  a  maxi- 
mum stress  in  any  given  member.  The  components  of  one  reaction  known,  the  stress  in  the 
member  is  found  by  a  single  equation  of  moments  of  the  forces  upon  one  side  of  the  section; 
or  if  a  web  member,  by  a  summation  of  the  components  of  all  the  forces  in  a  direction  normal 
to  the  arch  at  the  section  considered. 

If  the  arch  under  consideration  is  a  circular  one,  the  reactions  may  be  found  with  suffi- 
cient accuracy  in  all  ordinary  cases  by  using  instead,  a  parabolic  arch  of  the  same  span,  and 
whose  average  length  of  ordinate  is  the  same  as  that  of  the  given  circular  arch  ;  that  is,  one 
which  encloses  the  same  area  between  the  arch  and  the  line  joining  the  springing  points.  The 
rise  of  such  an  arch  is  readily  found,  remembering  that  the  area  enclosed  by  the  parabolic 
arch  is  equal  to  ^rc.  After  obtaining  the  reactions  the  stresses  should  be  found  for  the  actual 
arch. 

192.  Temperature  Stresses. — A  rise  of  temperature  of  /  degrees  above  the  normal, 
lengthens  each  element,  ds,  of  the  arch,  by  an  amount  equal  to  ieds,  where  e  is  the  coefficient 


212 


MODERN  FRAMED  STRUCTURES. 


of  expansion.    The  horizontal  component  of  this  increment  is  Uds  cos  z  or  tedx,  and  the  total 

change  in  length  of  s/>an  if  the  arch  were  free  to  move  would  be  equal  to        tedx  =  2cte. 

This  movement  is,  however,  resisted  by  horizontal  abutment  reactions,  and  stresses  are  thus 

developed  in  the  arch. 

Let  A'B,  Fig.  253,  be  the  normal  position 
of  the  lengthened  diXch,  and  AB  the  position  when 
sprung  or  kept  in  place  by  the  abutments, 
A' A  being  equal  to  2cte.  The  tangents  to  the 
two  curves  at  B  are  respectively  BT'  and  BT. 
The  movement  of  A'  to  A  may,  as  in  Art.  188, 
be  conceived  as  divided  into  two  parts ;  one  a 
turning  about  ^  as  a  centre  through  the  angle 
(5,  the  point  A'  coming  to  A",  and  the  other 
part,  that  due  to  bending,  the  end  of  the  arch  moving  from  A"  to  A.  As  in  Art.  188,  the 
horizontal  component  of  A' A"  may  be  put  equal  to  zero,  whence  the  horizontal  component 
of  A"A  =  A'A  =  2cte,  or,  from  eq.  (9), 


Fig.  253. 


•  Myds 


(22) 


or,  according  to  the  assumptions  of  Art.  189, 

I  n^c 

-^j        Mydx  =  2cte.  (23) 

Now  in  Fig.  254  the  moment,  M,  at  any 
point  n  due  to  two  equal  horizontal  reactions 
Hi,  is  Hty\  hence  xiHt  is  the  reaction  caused  by  a 
^  rise  of  temperature  of  t  degees,  eq.  (23)  becomes 


Fig.  254. 


Eh 
16 
is 


/■2C 
y^dx  ■=■  2cte. 


(24) 


Pzc  ID 

From  eq.  (15),  p.  210,  the  value  of  /  y''dx  is  equal  to  — rV;  hence,  substituting  in  (24) 
and  solving  for  Ht,  we  have 

iSEIJe 


8/^ 


(25) 


From  this  equation  the  value  of  Ht  can  be  computed  for  any  change  of  temperature  above  or 
below  the  temperature  for  which  the  arch  is  designed,  and  the  stresses  readily  found  in  all 
the  members  by  means  of  a  stress  diagram.  For  a  fall  of  temperature,  t  is  of  course  negative 
and  Ht  acts  outwardly. 

193.  Stresses  due  to  Change  of  Length  from  Thrust. — From  the  direct  thrust  due 
to  loads  or  changes  of  temperature,  each  element  of  the  arch  is  shortened  by  an  amount 

equal  to'^s,  where /"=  compression  per  unit  area;  ds  =  length  of  element ;  and  E  =  modulus 

of  elasticity.  This  shortening  due  to  compression  has  precisely  the  same  effect  as  a  fall  of 
temperature.  The  value  of  /due  to  any  particular  loading  or  to  a  change  of  temperature  is 
not  constant  along  the  arch,  but  it  is  nearly  so  and  may  be  so  considered  for  our  purposes. 

Taking  for /,  then,  an  average  value  along  the  arch  for  any  given  loading,  the  quantity"^  will 


ARCH  BRIDGES. 


replace  te  of  the  preceding  article.  Eq.  (25)  therefore  becomes,  if  is  the  horizontal  reac- 
tion necessary  to  resist  the  shortening  of  the  arch, 

 (26) 

the  minus  sign  indicating  that  acts  outwardly,  or  in  the  direction  of  Ht ,  for  a  fall  of  tem- 
perature. 

In  getting  stresses  it  will  be  convenient  to  assume  some  vaUie  of  H, ,  as  100,000  lbs.  for 
example,  and  with  this  value  find  the  stresses  in  all  the  members  by  means  of  a  stress  diagram, 
the  only  external  forces  acting  being  the  two  equal  horizontal  reactions.  Then  for  any  par- 
ticular member,  find  an  average  value  of  /by  getting  its  value  at  three  or  four  points  along  the 
arch,  when  loaded  so  as  to  produce  the  maximum  stress  in  the  member  in  question  ;  substitute 
this  average  value  in  eq.  (26)  and  determine  .  Then  multiply  the  stress  found  from  the 
diagram  by  this  value  of      and  divide  by  100,000. 

Average  values  of  /due  to  the  maximum  rise  and  fall  of  temperature  maybe  found  once 
for  all,  the  corresponding  values  of  computed,  and  these  added  to  the  values  of  as 
found  above. 

Stresses  due  to  the  shortening  of  the  arch  may  be  prevented  to  a  large  extent  by  con- 
structing it  a  little  longer  than  the  span,  and  springing  it  into  place.  Under  certain  loads,  then, 
the  arch  is  compressed  just  enough  to  make  its  length  normal,  and  the  stresses  due  to  shorten- 
ing are  zero.  This  initial  lengthening  of  the  span  by  a  given  amount,  J,  has  the  same  effect  as 
a  rise  of  temperature  which  lengthens  the  span  by  the  same  amount.  Hence,  in  eqs.  (24)  and 
(25),  if  we  substitute  J  for  2cU,  and  if  Hi  is  the  horizontal  thrust  due  to  the  initial  lengthen- 
ing, we  have 

i6cr' 

This  value  of  //^  is  to  be  added  to      of  eq.  (26). 

The  proper  value  of  z/  to  be  used  in  designing  the  arch  may  be  found  by  substituting  for 
Hi  in  eq.  (27)  an  average  value  of  H^  as  found  from  eq.  (26),  and  then  solving  for  //. 


(27) 


PARABOLIC  ARCH  WITH  FIXED  ENDS  ;  VARIABLE  MOMENT  OF  INERTIA. 

194.  The  Equilibrium  Polygon. — Fig.  255  represents  such  an  arch,  the  ends  at  A  and  B 
being  fixed  in  direction  as  well  as  position  by  the  abutments.    This  being  the  case,  there  will 


1 


Fig.  255. 

in  general  be  some  bending  moment  at  A  and  B,  and  the  equilibrium  polygon  for  a  load  P 
will  not  pass  through  these  two  points.  Letj,  be  the  ordinate  from  the  equilibrium  polygon 
to  that  abutment  farthest  from  the  load  F,  and     be  the  ordinate  to  the  other  abutment. 


214  MODERN  FRAMED  STRUCTURES. 

Let  be,  as  before,  the  ordinate  from  k,  the  intersection  of  the  two  segments  of  the  polygon, 
to  the  hne  AB.    The  unknowns  are  here  the  three  ordinates , ,  andj,. 

In  Art.  1 88  we  have  seen  that  for  an  arch  with  no  hinges  we  have  the  three  conditions, 
from  eqs.  (7),  (8),  and  (9), 


rMds  c 


-^  =  0,    and  1-^=0. 


Making  the  same  assumptions  as  in  Art.  189  regarding  E  and  /,  and  putting //"^  or 
—  2'),  for  M,  the  above  equations  become 

ydx  —  J*^  z'dx  =  o ;  (28) 

xydx  —  J*^  z'xdx  =  o;  (29) 

J  y'dx  —      z'ydx  =0.  »    .    ■  (30) 

The  values  of  these  integrals  will  now  be  derived. 

Equation  (28). — From  eq.  (14),  p.  209,  we  have  j/  =  ^(2^^  —  •^)-  Substituting  this  value 
in  the  first  term  of  (28)  and  integrating,  we  have 

S.  ■^'^•^^X  ^-^{2C  -  x)  dx  =  ^rc  {d) 

The  second  term  of  (28)  is  simply  the  area  of  the  two  trapezoids  ALkG  and  BMkG,  or 
£  z'dx=^-^^-^{c+b)+^-^{c-b)  {b) 

Subtracting  {d)  from  {b)  and  reducing,  we  have  from  (28) 

2^7„  +{c-\-  b)y,     {c  -  b)y,  -  %rc  =  o,   (3 1) 

Equation  (29). — Referring  to  Fig.  255  we  see  that  the  first  term  of  (29)  is  simply  the 
moment  of  the  area  between  the  parabola  ACB  and  the  line  AB,  about  the  axis  AY.  This 
area  is  equal  to  ^cr ;  hence 

xydx  =  ^cr  X  c  =  ^c'r  (c) 

The  second  term  of  (29)  is  likewise  the  moment  of  the  area  ALkMB  about  A  Y.  This 
area  may  be  divided  into  the  two  rectangles  ALTG  and  BMSG,  and  the  two  triangles  LkT 
and  SMk.    Writing  out  these  moments,  we  have 

£^  z'xdx  =  y^^-^  +ylc  -  b)\^2c  -  \{c  -  b)\ 

Subtracting  {c)  from        we  have  after  reduction 

2<3^  +  %o+(^  +  /5)>.  +  (^-'^)(5^  +  %,-8^V  =  o.      .   o   ,    .  (32) 
Equation  (30). — From  eq.  (15),  p.  210,  we  have 

X"/dx  =  \yc.    ............  (^) 


ARCH  BRIDGES.  215 
For  the  second  term  of  (30)  we  have,  from  A  to  G, 

and  if,  for  the  portion  from  G  to  B,  we  take  our  origin  at  B,  we  have  for  this  portion 

TX 

The  value  of  y  is,  for  both  cases,  equal  to      {2c  —  x)  as  before.    We  may  then,  as  in  Art. 

189,  substitute  the  first  value  of  z'  in  (30)  and  integrate  from  o  to  {c-\-b),  then  the  second 
value,  and  integrate  from  o  to  (c  —  b)  with  origin  at  B.    That  is, 

/'     = /^'?(^'  -     +  Tfi^"-)"^- + /"'?(^^  -  Ay-  +  T^^y-  (/) 

Performing  the  integrations  indicated,  collecting  terms,  combining  with  {e),  and  multiplying 
by        we  have  finally 

2c{^c^-b')y,^{cJrb)\lc-b)y,^{c-b)\ic^b)y,-^rc'=0.    .    .    .  (33) 

We  now  have  three  equations,  (31),  (32),  and  (33),  between  three  unknown  quantities, 
j„ ,  ,  and  y^ .  To  solve  these  equations,  multiply  (31)  by  {c  -\-  b)  and  combine  with  (32)  to 
eliminate  j/, ,  giving 

4^>.  +  ¥{c  —  %,  —  -jC^*^'-  bc)=o.    .    .  {g) 

Multiplying  (32)  by  (3^  —  b)  and  combining  with  (33),  we  have,  similarly, 

4^>o  +       —  b)j'i  —  4^1'^'  —  bc)=:0  (A) 

Subtracting  (g)  from  {/i)  and  solving  for     ,  we  have 

Substituting  in  (g)  or  {k),  we  obtain 

7.  =  !''  (35) 

And  finally  by  substituting  in  either  of  the  first  three  equations  we  have 

2   c  4- 

Eq.  (35)  shows  that  the  locus  of  k  is  a  horizontal  straight  line  at  a  distance  of  above 
AB.  The  value  of  in  eq.  (36)  varies  from  —  00  for  b  =  —  c  to  a.  value  of  -j\r  for  b  =  -j-  c. 
The  value  of  jj/,  varies  in  the  opposite  way. 

195.  Position  of  Loads  for  a  Maximum  Stress  in  any  Member. — All  the  reaction  lines, 
Lk,  constructed  for  loads  at  various  points  along  the  arch  are  tangent  to  some  one  curve ;  like- 


2l6 


MODERN  FRAMED  STRUCTURES. 


wise  for  the  lines  kM.  In  finding  position  of  loads,  it  will  be  convenient  to  construct  these  curves 
or  envelopes.  These  curves  are  hyperbolas,  and  their  equations  might  be  derived  ;  but  it  will 
be  as  convenient  to  construct  them  by  drawing  the  reaction  lines  Lk  and  Mk  for  several  posi- 

y  y 

tions  of  the  load,  or  for  several  values  of  b.    The  following  table  gives  values  of  ^  and  for 

various  values  of      which  will  enable  enough  lines  to  be  drawn  to  locate  the  curves  with 

c  ^ 

sufificient  accuracy. 


TABLE  OF  VALUES  OF  -    AND    -  FOR  VARIOUS  VALUES  OF  * 

r  r  c 


b 

-  0.8 

-  0.6 

+  0.2 

+  0-4 

+  0.6 

t 

—  I.O 

-  0.4 

—  0.2 

0.0 

+  0.8 

+  1.0 

y± 

r 

—  oo 

—  2.0 

—  0.667 

—  0.222 

±  0.0 

+  0.133 

-j-  0.222 

+  0.286 

+  0.333 

+  0.370 

+  0.400 

yi 

4-  0.40 

+  0.370 

+  0.333 

+  0.286 

+  0.222 

+  0-133 

±  0.0 

—  0.222 

—  0.667 

—  2.00 

  CO 

r 

In  Fig.  256,  ac  is  the  locus  of  k,  being  equal  X-o  \  de  is  the  envelope  of  the  lines  Lk 
and      is  the  envelope  of  the  lines  kM.    If  G  is  the  centre  of  moments  for  any  member  D£, 


Fig,  256. 


the  lines  Gk  and  Gk',  drawn  tangent  to  de  and  fg-  respectively,  will  determine  the  position  of 
the  loads  for  a  maximum  stress  of  either  kind  in  DE,  since  loads  between  k  and  k'  cause 
positive  moment  at  G,  while  all  other  loads  cause  negative  moment.  Likewise  a  line  drawn 
tangent  to  de  and  parallel  to  FG  and  D£,  or  towards  their  intersection,  will  determine  the 
position  of  loads  for  maximum  stress  of  either  kind  in  DG,  as  in  Art.  190. 

196.  Computation  of  Stresses. — The  stresses  can  be  obtained  with  sufificient  accuracy 
by  graphical  methods.  A  skeleton  diagram  of  the  arch  should  be  drawn  to  a  large  scale  and 
the  equilibrium  polygon  and  force  diagram  drawn  for  each  joint  load,  computing  j,  and  7, 
by  eqs.  (36)  and  (34).  The  stresses  are  then  found  in  each  member  due  to  each  load  by  the 
method  of  Art.  184. 

If  analytical  methods  are  preferred,  the  following  equations  will  enable  the  reactions  for 
any  load  to  be  computed : 


ARCH  BRIDGES. 


atf 


In  Fig.  255  we  have,  by  similar  triangles, 

Adding  and  putting  F,  +  F,  =  P,  we  have,  after  solving  for  H, 

H  =  . 

J.  -  7.  I    -  y% 

c-j-d       c  -  6 

Substituting  from  eqs.  (34),  (35),  and  (36)  and  reducing,  we  have 


•      •      •      a      •  • 


(37) 


^2  L 


32  L  -i  r 


By  substituting  in  (37)  and  reducing,  we  have 
and 


(38) 

(39) 
(40) 


By  means  of  these  equations  the  components  of  the  reactions  due  to  each  load  can  be 
computed  and  the  resulting  F's  added  to  get  the  total  V  due  to  those  loads  which  act 
together  to  cause  the  maximum  stress  in  any  member.  The  several  N's  may  also  be  added, 
and  the  line  of  action  of  the  resultant  H  be  found  by  multiplying  each  H  by  the  correspond- 
ing jj/,  or/,  and  dividing  the  sum  of  these  products  by  the  sum  of  the  H's.  The  components 
and  line  of  action  of  one  reaction  being  known,  the  stress  in  the  member  in  question  is 
readily  found. 

If  a  circular  arch  is  under  consideration,  the  reactions  may  be  found  by  a  similar  approxi- 
mation to  that  explained  in  Art.  191. 

197.  Temperature  Stresses. — For  an  arch  with  fixed  ends,  the  H/s  necessary  to  resist 
a  lengthening  of  the  span  due  to  a  rise  of 
temperature  are  not  applied  at  A  and  £,  but 
at  some  distance  orj/,  above,  similarly  to 
the  horizontal  forces  for  vertical  loads.  In 
this  case,  for  equilibrium,  j/^  =  h.  The 
ends  of  the  arch  at  A  and  B  being  fixed  in 
direction,  we  must  have,  from  eq.  (7),  when 

Mdx  =  o,  since  M 


 r—H, 


\h  X 

i  -^X3 

yjii 


Hly-h\ 


/2C  y^lC 
ydx  —  jf   hdx  —  o. 

From  eq.  {d)  of  Art.  194  we  have 


(41) 


C  ydx  = ' 


rc, 


and  we  have  aLo 

hdx  —  2ch ; 

hence  substituting  in  (41)  and  solving  for  h,  we  have 

^  =  f    a  constant. 


(42) 


2l8 


MODERN  FRAMED  STRUCTURES. 


Now  the  lengthening  of  the  span  due  to  a  rise  of  temperature  of  t  degrees  above  the 
normal  is  2cte,  and  we  have,  as  in  eq.  (23),  p.  212, 

^^£Mydx  =  2cte.   (43) 

From  Fig.  257  we  have,  for  the  moment  at  any  point  n, 

M  =  H,z  =  H^y  -  h). 

Substituting  in  (43),  we  have 

-^'"^-^  ~  ^'X  ^^^~\  =  2t/^.    .......    .  (44) 


16 

By  eq.  (i  5),  p.  2 10.      /    fdx  =  —  rV, 
^0  15 


5 


P  4 
Also  we  have  /    ydx  =  area  A  CB  =  -  rc. 

^0  3 
Substituting  in  (44)  and  solving  for  Hi ,  we  have 


4r' 


(45) 


Knowing  the  amount  and  line  of  action  of  H^,  the  resulting  stresses  may  be  computed  by 
moments  and  shears,  or  in  case  of  a  truss  the  stresses  are  more  readily  found  by  a  diagram, 
the  stresses  in  two  or  three  of  the  end  members  being  first  found  by  moments. 

198.  Stresses  due  to  Change  of  Length  from  Thrust. — In  this  case,  as  in  Art.  193, 
the  effect  of  the  shortening  of  the  arch  is  the  same  as  that  caused  by  a  fall  of  temperature  ; 

and,  as  in  that  case,  we  may  substitute     for  te  in  eq.  (45)  and  we  will  obtain  the  outward 

thrust,  H,,  necessary  to  resist  this  shortening.  Hence 

^s=-^   .  (46) 

The  line  of  action  of  this  thrust  is  evidently  the  same  as  for  since  eq.  (42)  was  derived 
independently  of  the  cause  of  the  thrust. 

The  stresses  due  to  change  of  length  are  found  as  in  Art.  193. 


DISTRIBUTION  OF  LOADS  OVER  REDUNDANT  MEMBERS. 


CHAPTER  XV. 

DEFLECTION  OF  FRAMED  STRUCTURES  AND  THE  DISTRIBUTION  OF  LOADS  OVER 

REDUNDANT  MEMBERS. 

199.  The  Deflection  of  a  Framed  Structure  for  any  given  loading  can  be  as  rigidly 

computed  as  the  stresses  in  the  members  can  be  found.  Although  it  would  seem  to  be  self- 
evident  that  the  extension  or  shortening  of  any  main  truss  member  must  contribute  somewhat 
to  the  deflection  as  a  whole,  it  has  long  been  customary  to  state  that  the  deflection  is  almost 
wholly  due  to  the  stresses  and  strains  obtaining  in  the  chords,  and  but  little  attention  has 
been  given  to  the  web  members  in  this  connection.  Probably  Stoney's  two-volume  work  on 
"  Theory  of  Strains  in  Girders"  (2d  ed.,  London,  1869)  has  stated  this  position  most 
emphatically.  Indeed  he  has  placed  as  the  frontispiece  to  the  first  volume  what  purports  to 
be  a  graphical  proof  of  the  proposition.    He  says : 

"At  first  sight  it  may  be  thought  that  the  web  of  the  plate  girder,  or  the  braced  web  of  the  latticed 
girder,  will  seriously  affect  the  amount  of  the  deflection  curve ;  but  it  can  be  readily  shown  by  carefully  con- 
structed diagrams,  in  which  the  alterations  of  length  due  to  the  load  are  drawn  to  a  highly  exaggerated  scale, 
that  the  construction  of  the  web  has  scarcely  any  influence  on  the  curvature." 

He  then  constructs  a  drawing  showing  this  effect  on  a  very  shallow  Warren  girder,  and 
seems  to  prove  his  proposition.    He  says  of  them  : 

"These  diagrams  give  very  interesting  results;  they  show  that  the  curvature  of  flanged  girders  is  prac- 
tically independent  of  change  of  form  in  the  web  and  almost  entirely  due  to  the  shortening  of  the  upper, 
and  the  elongation  of  the  lower,  flange ;  and  a  further  inference  may  be  derived  from  them,  viz.,  that 
deflection  is  practically  unaffected  by  the  nature  of  the  web,  whether  it  be  formed  of  plates  or  lattice  bars."  * 

These  conclusions  prove  to  be  erroneous  and  have  been  grossly  misleading.  In  actual 
structures  deflections  computed  from  chord  strains  only  have  been  found  to  be  about  half 
as  great  as  those  actually  observed  under  loads,  and  the  difference  has  been  set  down  as 
another  illustration  of  the  "  universal  discrepancy  between  theory  and  practice." 

In  what  follows  a  method  will  be  given  for  accurately  computing  the  deflection  of  any 
point  in  any  framed  structure,  under  any  given  loading,  and  it  will  also  be  shown  that  this 
information  may  be  used  to  determine  stresses  in  redundant  members,  otherwise  indetermi- 
nate.f 

200.  Fundamental  Propositions. — Three  propositions  will  first  be  stated  and  proved ; 
Proposition  I.    The  external  work  of  distortion  of  a  fra7ned  structure  by  a  load  is  equal 

'to  the  internal  work  of  resistance. 


*  Vol.  I.,  2d  ed.,  pp.  141,  142. 

t  The  remaining  portion  of  this  chapter  mostly  appeared  as  a  Paper  by  Prof.  Johnson  before  the  Engineers'  Club 
of  St.  Louis,  and  was  published  in  the  Jour.  Assoc.  Eng.  Socs.,  for  May,  1890. 


930 


MODERN  FRAMED  STRUCTURES. 


Proposition  II.  The  deflection  of  any  point  in  any  framed  structure  subjected  to  any 
given  load  is  given  by  the  formula 

D  =  2^.*  W 

where  D  =  deflection  of  point  under  consideration  ; 

/  =  stress  per  square  inch  in  any  member  for  the  given  load ; 
/  =  length  of  any  member  ; 
E  =■  modulus  of  elasticity  of  any  member; 
u  =  factor  of  reduction  ; 
2  =  sign  of  summation. 

That  is  to  say,  the  quantity  ^^\s  computed  for  every  member  of  the  truss  and  the  algebraic 

sum  taken  as  the  total  deflection  of  the  point. 

Proposition  III.  When  there  are  two  or  more  paths  over  which  a  load  may  travel  to  reach 
the  support,  the  load  divides  itself  among  the  several  paths  strictly  in  proportion  to  the  rigidities 
of  the  paths. 

The  relative  rigidities  of  the  paths  are  indicated  by  the  relative  loads  required  to  produce 
a  given  deflection,  or  they  are  inversely  as  the  deflections  produced  by  a  given  load  which  is 
required  to  pass  wholly  over  each  path  in  succession.  Having  established  this  proposition 
and  computed  the  rigidities  of  the  paths  by  Prop.  II,  we  can  write  enough  equations  of 
condition  to  enable  us  to  solve  for  any  number  of  redundant  members.  This  proposition 
applies  to  all  structures,  whether  framed  or  composed  of  masonry,  as  in  the  case  of  a  curved 
masonry  dam. 


PROOF  OF  propositions. 

Proposition  I  hardly  requires  proof.  It  is  simply  expressing  for  the  work  or  energy 
spent  on  the  structure  that  which  we  take  for  granted  as  to  the  force  coming  upon  it ;  that  is 
to  say,  action  and  reaction  are  equal.  The  external  work  of  distortion  is  the  product  of  the 
load  into  the  deflection  of  the  loaded  point  divided  by  two.  The  internal  work  of  resistance 
is  the  sum  of  the  products  of  the  stress  produced  in  each  member  by  its  distortion,  divided  by 
two. 

Thus  if  W^is  the  external  load, 

D  the  deflection  of  the  loaded  point, 

Pthe  stress  produced  in  any  member, 

z  the  distortion  of  any  member  due  to  the  stress  P, 

then  we  have 

WD  Pz 

—  =     ,  (i) 

2  2  ^  ' 

Pz 

where  the  second  member  represents  the  algebraic  sum  of  the  quantities  —  computed  for  all 

the  members  of  the  truss  for  the  load  W. 

From  the  above  it  would  appear  that  the  total  deflection  D  at  the  loaded  point  is  made  up 
of  as  many  parts  as  there  are  members  in  the  truss,  each  one  contributing  its  portion,  cor- 

Pz 

responding  to  the  expression  — •  for  that  member.    If  we  represent  that  portion  of  D  resulting 


*  Conveniently  remembered  as  the  "pull  over  E"  formula. 


DISTRIBUTION  OF  LOADS  OVER  REDUNDANT  MEMBERS. 


221 


from  the  distortion  in  one  member  by  d,  then  we  may  say  that  the  work  done  at  the  loaded 
point  corresponding  to  the  work  of  resistance  in  any  particular  member  is 

Wd     Pz  P  . 

• — •  —  — ,    or  d  —  tTt^i  •••    •••••    •••  \2) 

2  2  W 

In  the  above  discussion  we  have  considered  the  truss  as  loaded  only  at  the  point  whose 
deflection  is  under  consideration,  but  by  so  doing  we  have  obtained  in  (2)  a  relation  between 
the  distortion  of  any  particular  member  and  the  vertical  deflection  of  our  loaded  point,  which 

p 

is  quite  independent  of  W,  since  the  quantity  jj/is  a  constant  ratio  for  any      whatsoever.  In 

other  words,  equation  (2)  is  the  kinematical  relation  between  the  distortion  of  any  member 

P 

and  the  deflection  at  the  given  point  produced  by  that  distortion.    The  ratio       may  be 

found  by  assuming  any  W  dX  our  given  point  and  finding  P  analytically  or  graphically  for  the 

p 

particular  member.  Let  ^  =  where  u  may  be  defined  as  the  stress  in  the  particular 
member  when      =  i  lb.  Then 

d  =  uz. 

Now  suppose  the  truss  loaded  in  any  manner  whatsoever  and  let  the  resulting  distortion  of 
a  particular  member  be  z'  and  its  unit  stress  =■  p.    Then  d  =  uz'  where    is  a  known  quantity. 

But  we  know  that  z'  =      where  p  has  been  found  analytically  or  graphically  for  that  particular 

II 

loading  ;  therefore 

 ^3) 

or  the  deflection  d  at  any  point  due  to  the  distortion  ^  of  any  one  member  is  equal  to  that 

distortion  multiplied  by  a  number  71,  numerically  equal  to  the  stress  in  the  member  caused  by 
placing  I  lb.  at  the  point  in  question. 

The  student  should  guard  himself  here  against  the  too  hasty  conclusion  that  the  above 
demonstration  applies  only  to  deflections  of  a  loaded  point.  The  relation  between  the  deflec- 
tion of  any  point  and  the  deformation  of  any  member,  from  any  cause,  is  shown  above  to  be 
the  same  relation  as  exists  between  a  load  placed  at  tJiat  point,  and  the  resulting  stress  in  that 
member.  This  latter  relation  being  readily  found,  it  is  used  for  the  kinematical  relation 
sought. 

The  total  deflection  of  the  point  is  therefore  the  sum  of  all  its  parts,  or 

^=2^^.   (4) 

Hence  follows  Proposition  II. 

When  the  point  whose  deflection  is  desired  is  the  middle  point  of  a  truss  both  /  and  u 
will  always  have  the  same  sign  for  all  members  when  the  bridge  is  fully  loaded,  and  hence 
their  product  will  always  be  positive.  If  any  other  pomt  be  chosen,  or  if  the  load  be  an 
unsymmetrical  one,  this  product  may  be  negative  in  a  few  cases,  where  the  algebraic  sum 
must  be  taken. 

In  applying  this  in  practice  we  always  know  p,  the  unit  stress  in  every  member,  for  the 
assumed  load  on  the  structure,  also  its  length  /,  and  its  modulus  of  elasticity  E.  It  remains 
therefore  only  to  find  u  for  every  member.  Since  this  is  a  pure  ratio,  equal  to  the  stress  in 
that  member  for  one  pound  placed  at  the  point,  we  simply  put  i  lb.  at  the  point  whose 


222 


MODERN  FRAMED  STRUCTURES. 


deflection  is  desired  and  find  the  resulting  stress  in  every  member,  either  analytically  or 
graphically.    This,  then,  is  to  be  considered  as  an  abstract  quantity,  being  in  fact  a  pure  ratio, 

and  the  factor  of  reduction  by  which  the  distortion  ^  of  each  member  is  reduced  to  the  re- 

E 

suiting  deflection  of  the  point.  This  point  would  usually  be  the  end  of  a  cantilevered  arm,  as 
in  a  swing  bridge,  or  the  middle  point  of  a  truss  supported  at  the  ends.  It  may,  however,  be 
any  point,  and  the  truss  may  be  of  any  design,  so  long  as  the  stresses  are  all  direct  tension 
and  compression.  The  formula  can  take  no  account  of  any  bending  stresses  nor  of  any  lost 
motion  at  joints.  This  formula  may  be  applied  to  all  kinds  of  trussed  forms.  The  load  on 
the  truss  may  be  any  assumed  load  whatsoever  for  which  the  unit  stresses,  are  computed. 
But  the  one-pound  load,  for  finding  u  for  each  member,  must  be  put  at  the  point  whose 
deflection  is  desired.* 

The  truth  of  Proposition  III  is  also  nearly  self-evident.  If  the  paths  be  conceived  as 
india-rubber,  all  taut  and  ready  to  act  in  resisting  distortion,  then  for  a  given  distortion  the 
load  will  divide  itself  over  the  paths  in  proportion  to  their  several  resistances  to  distortion. 
But  the  degree  of  resistance  to  a  given  distortion  is  a  measure  of  the  rigidity  of  the  body. 
Hence  we  may  say  the  load  divides  itself  among  the  paths  in  proportion  to  their  respective 
rigidities. 

201.  Deflection  Formula  for  a  Pratt  Truss.— For  approximate  or  working  values  of 

the  deflection  of  any  style  of  truss  we  may  obtain 
a  formula  which  is  readily  evaluated,  provided 
we  may  assume  some  average  values  for  in- 
tensities of  the  tensile  and  compressive  stresses. 
Thus  for  a  Pratt  truss,  single  intersection,  of  an 
even  number  of  panels,  we  may  use  for  the 
mean  tensile  stress  and  for  the  mean  com- 
pressive stress,  E  for  the  modulus  of  elasticity, 
d  for  the  panel  length,  h  for  the  height  of  truss, 


Id 

Values  of  u 

Fig.  258. 


«  for  the  number  of  panels,  and  obtain  (referring  to  Fig.  258) 


For  the  Upper  Chord 

1 

For  the  Lower  Chord  *« 
For  the  Web  Tension  Members  *• 
For  the  Hip  Struts  «« 
For  the  Verticals  " 


pul 


2^  ^ 


8£^(«  +  4)(«-2); 
=  ^[«(«-2)+8]; 


•  (5) 


*  This  elegant  proposition  in  framed  structures  was  first  published  in  America  by  Prof.  Geo.  F.  Swain  in  the  Journal 
of  the  Franklin  Institute,  April,  May,  and  June  1883.  The  proposition  was  first  given  by  Lame,  and  afterwards  amplified 
and  its  application  extended  by  Maxwell  and  Jeiikin,  in  England,  and  by  Mohr  and  Winkler,  in  Germany.  In  Prof. 
Swain's  treatment  the  proposition  is  based  on  the  principle  of  virtual  velocities,  and  he  uses  it  for  finding  the  stresses 
in  various  styles  of  trusses,  particularly  arches  and  continuous  girders.  The  proof  given  above  based  on  the  equality 
between  the  external  work  of  deformation  and  the  internal  work  of  resistance  is  much  simpler  in  its  conception  and 
shorter  in  its  method.  The  proposition  may  also  be  proved  by  the  principle  of  least  work.  The  most  frequent  use  for 
this  proposition  probably  occurs  in  connection  with  the  deflection  of  structures  rather  than  in  the  computation  of  stresses. 

After  the  distortions  of  the  several  members  have  been  found,  the  change  of  position  of  every  joint  in  the  structure 
may  be  obtained  graphically  by  means  of  Williot  diagrams,  which  are  themselves  an  elegant  application  of  the  graphical 
method.  See  a  paper  on  The  Distortion  of  a  Framed  Structure  Graphically  Treated  by  David  Molitor  in  The  Journal 
of  the  Association  of  Engineering  Societies!,  June  iSg(.     By  this  method  both  the  horizontal  and  the  vertical  displac*- 


DISTRIBUTION  OF  LOADS  OVER  REDUNDANT  MEMBERS. 


Whence  for  the  whole  truss  the  total  deflection  of  the  middle  point  for  a  full  load  is 

2>  =  2^  =  ^%«  +  2)^+(«-2)^-]  (6) 

where  pi  —  average  unit  stress  of  tension  members; 

=  average  unit  stress  of  compression  members ; 
E  =  modulus  of  elasticity  for  all  members ; 
h  =  height  of  truss  in  inches  ; 
d  =  panel  length  in  inches  ; 
n  =  number  of  panels  in  bridge. 
It  will  be  noticed  that  in  this  case  there  is  nothing  to  sum  but  ti  for  each  member.  a*» 
grouped  above  in  eqs.  (5), /,  and  E  being  constant  for  all  the  members  of  a  group.  Also 
for  such  members  as  give  a  value  of  u  =  o,  as  for  the  middle  vertical  and  the  end  hanger,  they 
are  of  course  omitted,  or  rather  count  for  nothing  in  the  summation.    This  means  that  these 
two  members  do  not  in  any  way  contribute  to  the  deflection  of  the  middle  point. 

Numerical  Example. — Take  a  Pratt  truss,  200  feet  span,  of  twelve  panels,  with  a  height  of  400  inches, 
0/  33  ft.  4  in.,  the  panel  length  being  200  inches.  If  the  average  maximum  tensile  stress  for  both  dead  and 
live  load  be  taken  as  10,000  lbs.  per  square  inch  and  the  average  compressive  stress  as  7000  lbs.  per  square 
inch,  the  total  deflection  is  readily  found  from  eq.  (6)  to  be  2.49  inches. 

202.  Relative  Deflection  from  Web  and  Chord  Stresses. — By  adding  the  deflection 
Increments  due  to  web  members  and  those  due  to  chord  members,  we  may  obtain, 


For  Chord,  2^  =  ^[(«  («  -  2)  +  8)  A  +  («  +  4)  («  -  2)A]  ; 
For  Web,       «    =  ^[(«  -  2)  {k'  +  d')Pi  +  ((«  -  2)/t*  +  2^/*)/,]. 


...  (7) 


If  we  should  assume  that  the  average  stress  in  the  compression  members  is  0.7  that  in 
the  tension  members  or  p^  =  o.jpi ,  we  may  write. 


For  Chords,  =  ^l"^'"^^*  ~  ^'^^  +  ^'^  ' 

For  Web,        «    =  ^[(i.7«  -  I-AW  +  («  -  0.6K']. 


(8) 


Whence 


^  ^     .     .  .         (6.8«  —  i3.6)f-,)  +  4»  —  2.4 

Deflection  from  web        ^  ^  '\di^^  ^ 

-  (9) 


Deflection  from  chords  i.'jt^  —  o.6n  +  2.4 

A 
d 


h 

This  ratio  increases  as  -5  increases,  and  decreases  as  «,  the  number  of  panels,  or  length  of 


bridge,  increases. 

For  a  Pratt  truss  bridge  of  200  feet  span,  of  ten  panels,  and  a  height  of  30  feet,  this 
fraction  becomes  |^  or  96.4^^ 

That  is  to  say,  for  such  a  span  and  for  the  assumptions  made,  the  deflection  from  web 
distortion  is  about  equal  to  that  from  chord  distortion. 

A  Whipple  truss  is  simply  a  pair  of  Pratt  trusses  joined  into  a  double  intersection  system, 
and  hence  the  deflection  of  the  combination  is  to  be  computed  by  taking  one  of  the  systems 
alone. 

 ^  ^  I  

ments  of  every  joint  in  the  structure  are  obtained  by  a  single  graphical  construction.  If,  however,  only  the  vertical 
deflection  of  a  single  point  in  a  structure  is  desired,  the  graphical  method  offers  no  advantages  over  the  algebraic  method 
here  given. 


224 


MODERN  FRAMED  STRUCTURES. 


Thus  for  such  a  bridge,. 400  feet  long,  with  twenty  panels  of  20  feet  each,  the  panel  length 
for  one  system  would  be  40  feet.  Let  the  height  be  60  feet,  whence  we  would  have  «  =  10, 
d  =  40,  A  =  60. 

For  this  case  equation  (9)  would  give  exactly  the  same  as  before,  since  n  is  the  same  and 
^  gives  the  same  ratio. 

For  the  case  when  ^  is  large  and  n  small,  as  for  a  bridge,  say,  of  eight  panels,  of  15  feet 
each,  with  a  height  of  25  feet,  we  should  find 

Deflection  from  web       142.8  _ 
Deflection  from  chords"  106.6  ~"  ^'^^ 

That  is,  for  such  a  case  the  deflection  from  the  web  system  is  1.34  times  that  from  the  chord 
system.  On  the  other  hand,  if  the  height  is  about  equal  to  the  panel  length,  and  the  number 
of  panels  is  large,  as,  for  instance,  n  =  12  and  k  =  d,  then  we  would  find  that 

Deflection  from  web  28.4 


Deflection  from  chords  60 


0.47. 


Or,  in  this  case,  the  deflection  from  web  would  be  only  about  half  that  from  the  chords,  or 
one-third  the  total  deflection  of  the  bridge. 

In  general,  it  may  be  said  that  the  deflection  of  a  truss  bridge  from  the  web  stresses  is 
about  equal  to  that  from  the  chord  stresses.*  This  is  directly  contrary  to  the  usually  re- 
ceived opinions  of  engineers,  who  generally  assume  that  the  deflection  from  web  stresses  is 
relatively  insignificant.  It  is  probable  that  Mr.  Stoney  is  largely  responsible  for  this  generally 
accredited  opinion,  as  explained  above. 

203.  The  Effect  on  the  Deflection  of  changing  the  Height  of  the  Truss,  everything 
else  remaining  the  same. — We  may  differentiate  equation  (6)  for  h  variable  and  find 


dh 


Pc  ■\-ptr      {n  +  2)nd 


2E 


[(«  —  2)4^' 


+  («-2)], 
(«  +  2)«^»]  (10) 


This  is  the  change  in  the  deflection  for  a  change  of  one  inch  in  the  height  of  the  truss. 

Putting  this  quantity  equal  to  zero,  and  solving  for  k,  we  find  the  height  of  truss  which 
will  give  a  minimum  deflection  to  be 

n-\-2  nd* 


k 


_d  ln-\- 
~  2\  n  — 


(") 


for  a  minimum  deflection. 

From  eq.  (11)  we  find  that  for  a  minimum  deflection,  or  for  a  maximum  stiffness,  for 
given  working  unit  stresses,  the  height  of  this  stiffest  truss  has  the  following  values : 
TABLE  OF  HEIGHT  OF  PRATT  TRUSSES  OF  MAXIMUM  STIFFNESS. 


i.73</ 

I.i2d 

i.94(/ 

2.0'id 

2. ltd 

2.2-]d 

2.371/ 

2.42i/ 

Mo.  of  panels. . 

4 

6 

8 

10 

12 

14 

16 

18 

20 

Length 
Height 

2.3 

3-4 

4-3 

5.3 

5.9 

6.7 

7-1 

7-7 

8.3 

*  See  numerical  example  in  Art.  207. 


DISTRIBUTION  OF  LOADS  OVER  REDUNDANT  MEMBERS.  225 


Or  we  may  say  that  for  maximum  stiffness  for  a  given  an^.ount  of  material  the  height  of  the 
truss  should  vary  between  i|  and  2\  times  the  panel  length  and  from  0.43  to  0.12  of  the 
length  of  the  span,  as  the  number  of  panels  varies  from  4  to  20.  These  heights  being  very 
nearly  those  used  in  the  present  practice  of  bridge  designing,  there  is  no  new  moral  to  be 
pointed  from  this  conclusion.* 

From  eq.  (i  i)  it  may  be  seen  that  for  7t  =  2,  h  =  .  But  by  consulting  the  figure  it  will 
be  seen  that  the  design  is  not  adapted  to  a  truss  of  two  panels. 

The  objection  to  any  general  formula  for  deflection  of  any  form  of  truss  is  that  it  neces- 
sarily involves  taking  average  values  for  the  tensile  and  compressive  stresses,  pi  and  p^.  These 
can,  of  course,  only  be  taken  approximately ;  but  if  such  averages  be  once  computed  for  a  few 
spans  of  different  lengths,  they  would  probably  be  found  to  be  nearly  constant  for  given  maxi- 
mum  working  stresses.  Besides,  the  assumed  working  loads  are  seldom  or  never  the  actual 
loads,  and  hence  this  new  assumption  of  average  working  unit  stresses  is  probably  as  justifiable 
as  the  other  assumptions  that  must  be  made  to  solve  at  all.  It  may  be  remarked,  also,  that 
the  equation  for  height  giving  minimum  deflections  for  given  fibre  stresses,  (11),  is  not  a  func- 
tion of  those  assumed  stresses.  That  is,  the  height  for  maximum  rigidity  for  any  unit 
stresses,  provided  these  stresses  remain  constant  for  varying  heights. 

204.  Inelastic  Deflection. — It  must  also  be  understood  that  neither  the  general  equation, 
(4),  nor  the  particular  one  for  a  Pratt  truss,  (6),  makes  any  provision  for  the  inelastic  deflection 
due  to  any  slack  at  the  joints  from  pin-holes  being  larger  than  the  pin.  Since  this  slack  would 
be  only  one-half  the  difference  between  diameter  of  hole  and  that  of  pin  at  each  end  of  the 
member,  probably  an  average  value  of  0.02  inch  would  be  about  right  for  every  member  of 
a  well-constructed  bridge.  Now  this  affects  the  web  as  well  as  the  chord  members,  and  it  is 
equivalent  to  lengthening  every  tension  member  and  shortening  every  compression  member, 
except  those  in  the  top  chord,  by  this  amount.  The  top  chord  would  be  considered  one 
member  from  end  to  end.  The  effect  of  such  lengthening  or  shortening  of  any  member  on 
the  deflection  is  given  by  the  same  ratio  u,  so  that  the  deflection  caused  by  this  slack  on  any 
member  is  o.02u,  using  the  u  for  that  member.  In  other  words,  the  total  inelastic  deflection 
from  joints  would  be 

Inelastic  Deflection  =  D'  =  o.022u,  (12) 

remembering  to  take  but  one  u  for  the  top  chord,  and  that  one  for  the  end  panel.  If  the  top 
chord  is  cut  at  every  panel  joint  and  the  sections  rest  on  pins,  which  is  seldom  the  case,  then 
all  the  ?/'s  must  be  summed. 

For  a  Pratt  truss,  of  an  even  number  of  panels,  with  top  chord  provided  with  riveted 
cover  plates  and  planed  joints,  we  have 


Inelastic  Deflection  =  D' =  o.022u  =  —L-       —  n s]d -\-  2{n  -  4)/i  +  2«  (13) 


For  the  Pratt  truss,  used  in  the  above  example,  where  «  =  12,  /t  =  400,  and  =  200, 
being  200  feet  long,  we  find  the  inelastic  deflection  to  be  equal  to  0.494  inch.  For  the  condi- 
tions named,  therefore,  the  total  deflection  of  this  truss  would  be  2.49 -[-0.49  =  2.98  inches,  or 
say  3  inches. 

This  is,  of  course,  the  total  deflection  due  to  both  dead  and  live  load,  when  the  maximum 
load  is  on,  or  it  is  the  amount  by  which  the  bridge  should  be  cambered  to  bring  it  horizontal 
under  its  maximum  working  load. 


*  Bender  has  shown  that  the  truss  having  maximum  stiffness  is  also  the  lightest,  for  given  unit  stresses.  PrineipUs 
iff  Economy  in  the  Design  of  Metallic  Bridges,  p.  23. 


226 


MODERN  FRAMED  STRUCTURES. 


205.  Camber  of  Bridges. — A  bridge  is  cambered  partly  for  appearance  and  partly  that 

the  top  chord  joints  may  come  to  a  square  bearing  when  the  maximum  load  is  on.  It  of 
course  adds  nothing  to  the  strength  of  the  bridge. 

The  camber  should  equal  the  total  deflection,  being  the  elastic  deflection  due  to  the 
maximum  dead  and  live  loads  and  the  inelastic  deflection  due  to  lost  motion  at  the  joints. 

The  general  formulas  for  these  two  kinds  of  deflections  are  (4)  and  (12),  and  for  a  Pratt 
;i  uss  they  are  given  by  equations  (6)  and  (13)  respectively. 

A  Whipple  or  any  double  intersection  truss  with  parallel  chords  may  be  considered  as 
two  Pratt  trusses  which  must  deflect  together;  and  hence  equations  (6)  and  (13)  may  be  applied 
to  them,  being  careful  to  take  n  and  d  for  one  system  only.  That  is,  n  would  equal  one-half 
the  total  number  of  actual  panels  and  d  would  equal  the  length  of  two  panels  of  the  double 
system. 

There  is  no  question  but  that  present  practice  of  allowing  the  upper  chord  to  spread  at 
top  where  the  sections  join,  relying  on  its  closing  up  when  the  maximum  load  is  on,  is  a  poor 
one.  If  the  cover  plates  are  thin  and  the  number  of  rivets  few,  then  these  joints  do  actually 
open  and  close  at  top  for  every  passing  train.  If  heavy  cover  plates  on  both  top  and  sides 
and  a  sufficient  number  of  rivets  are  used,  then  as  this  joint  is  riveted  up,  so  it  will 
remain.  In  this  case  it  should  be  riveted  up  in  a  horizontal  position,  and  then  cambered  up 
to  its  final  position  by  bending  it  as  a  whole  from  end  to  end.  This  can  readily  be  done  if  the 
chord  be  riveted  up  before  the  bridge  is  swung.  It  would  have  to  be  supported  horizontally 
and  all  pins  inserted  except  one  at  bottom.  After  riveting  up  the  top  chord  the  truss  could 
be  jacked  up  and  the  last  joint  closed.  The  upper  chord  will  then  bend  bodily  from  end  to 
end  and  there  will  be  no  movement  of  the  splice.  One  can  see  that  the  upper  chord  of  a 
200-foot  span  will  readily  bend  3  inches,  if  this  bending  is  continuous,  without  serious 
damage ;  whereas,  if  it  had  all  to  occur  at,  say,  five  or  six  joints,  it  would  be  a  source  of 
weakness. 

In  case  the  ends  of  the  channels  should  be  cut  square  so  as  to  come  to  a  full  bearing  for 
the  maximum  load,  but  riveted  up  with  the  camber  in,  then  for  all  ordinary  loads  the  stress  is 
all  thrown  on  the  bottom  flanges  of  the  chord  channels.  It  may  be  asserted  that  no  serious 
damage  would  result,  but  it  is  at  best  a  very  unskilful  way  of  carrying  this  stress  across  the 
spliced  section. 

206.  Errors  caused  by  Neglecting  Deflections  due  to  Web  Distortions.— In  all 

computations  of  stresses  in  metallic  structures  based  on  the  deflections,  or  distortions,  of  the 
structure  from  internal  stresses,  the  ordinary  formulae  give  erroneous  results. 

Thus  in  computing  the  stresses  in  a  continuous  girder  caused  by  the  settlement  of  its 
supports,  or  the  temperature  stresses  in  an  arch  between  fixed  abutments,  or  the  stresses  in  a 
stiffening  truss  of  a  suspension-bridge,  the  stresses  are  in  all  these  cases  functions  of  certain 
assumed  or  computed  distortions.  These  distortions  are  always  assumed  to  result  in  certain 
bending  moments  only,  and  to  be  wholly  provided  for  by  the  strains  of  the  chords.  No 
assistance  in  accommodating  this  distortion  is  credited  to  the  web  members.  By  whatever 
proportion  this  distortion  is  absorbed  by  the  web  members,  the  stresses  in  the  chord  members 
are  reduced  below  those  now  computed  for  them. 

In  the  case  of  metallic  arches,  and  stiffening  trusses  on  suspension-bridges,  the  distortion 
absorbed  by  the  web  is  small  because  the  trusses  are  shallow  as  compared  to  their  length. 

The  relative  absorption  by  the  web  of  the  horizontal  distortion  of  the  St.  Louis  arches 
due  to  a  change  of  temperature  is  only  about  one-sixth  of  the  total  amount.  In  the  compu- 
tations it  was  all  assumed  to  go  into  the  chords,  or  tubes. 

In  the  case  of  a  continuous  girder,  however,  the  depth  would  be  great  in  proportion  to  the 
span,  and  here  the  computations  of  stresses  due  to  a  settlement  of  supports  (not  supports  out 
of  level  as  usually  stated,  for  if  the  bridge  be  built  to  rest  on  such  supports  the  formulae  apply) 


DISTRIBUTION  OF  LOADS  OVER  REDUNDANT  MEMBERS.  22j 


should  take  account  of  the  deflection  which  may  be  attributed  to  the  web  system.  Otherwise 
the  computed  stresses  in  the  chords,  for  this  case,  would  be  about  twice  their  actual  amount. 

207.  Numerical  Computation  of  Deflection. — The  truss  shown  in  Fig.  259  is  one  from 
a  highway  bridge  on  Twenty-first  Street,  St.  Louis,  erected  in  1892.  The  following  is  a 
tabular  computation  of  the  deflection  of  this  bridge  under  its  full  live  load  only.  The  deflec- 
tion is  found  for  the  centre  (/"),  and  hence  the  i-lb.  load  is  placed  at  this  point  for  computing 


f 

K 

c 

\ 

\ 

/  i 

/ 

/ 

1 

a       b        c        d  e 

/        a       k        k  I 

Fig.  259. 


u.  Two  Maxwell  diagrams  gave  both  u  and  the  total  stress  in  the  members  for  full  load. 
Since  both  the  "loading  and  the  deflection  point  are  symmetrical,  we  may  find  the  values  for 
one-half  only  of  the  truss  and  then  multiply  the  final  result  by  two  for  the  corresponding 
members  on  the  other  half  of  the  truss. 


COMPUTATION  OF  DEFLECTION  AT  CENTRE  OF  TRUSS. 


Member. 

Length  in 
Inches. 

Stress  from  Live 
Load. 

Area  of  Sec- 
tion inSquare 
Inches. 

Stress  per  Square 
Inch.  /. 

E' 

Contribution  to  Truss 

Deflection  =  i-=-. 

S 

AB 
BC 
CD 
DE 
EF 
be 
cd 
de 

247 
246 
245 
244 

244 
244 
244 
244 
244 

+  169,000 
-j-  274,000 
+  337.000 
-j-  364,000 

4-  377.000 

—  165,000 

—  272,000 

—  333,000 

—  361,000 

60 
60 

77 
77 
77 
315 

49 

61 .2 
69. 1 

-\-  2.800 
+  4,570 
+  4.380 
+  4,730 
+  4,890 

-  5.240 

-  5,550 

-  5,440 

-  5,220 

+  .0247 
+  .0401 
+  0383 
+  .0381 
-j-  .0426 

—  -0457 

-  .0483 

—  -0474 

-  -0455 

+  0.38 
+  .69 

+  96 
-[-  1 . 21 

+  i-5> 

-  0.37 

-  <  .68 

-  -95 

-  1.20 

in. 

4-  0.0094 
+  -0277 
-1-  .0368 
-f  .0461 
+  -0643 
-j-  .0169 
-1-  .0328 

4-  -0450 
+  .0546 

Half 

sum 

=  +  0.3336 
2 

Total  Deflection  from  the  distortion  of  Chords 

=     0.6672  in. 

Ab* 

Be 
Cd 
De 

Ef 
Bb 
Cc 
Dd 
Ee 

377 
408 

435 
458 
476 
327 
360 
387 
408 

—  257,000 

—  228,000 

—  110,000 

—  53,000 

—  31,000 
+  148,000 
.-(-  93.000 
-j-  41,000 

+  7,800 

49 
33 
24 
15 

II. 2 
36.5 
24-5 
22.5 
22.5 

-  5,240 

-  6,910 

-  4,580 

-  3,530 

-  2,770 
+  4,050 
-j-  3,800 
+  "1,820 
+  350 

—  .0706 

—  . 1007 

—  .0712 

—  -0577 

—  .0471 

+  -0473 
+  .0489 
+  .0252 
-1-  .0051 

-  0.58 

-  -51 

-  -49 

-  -47 

-  .60 
+  -44 
+  41 
+  .41 
+  -40 

+  .0409 
-j-  -0514 
+  -0349 
+  -0271 
-1-  .0283 
4-  .0208 
-|-  .0200 
-|-  .0103 
-j-  .0020 

+  .2357 
2 


Total  Deflection  from  distortion  of  Web  =  +O.4714  in. 

Total  Elastic  Deflection  of  Truss  for  Live  Load  =  1. 1386  in. 
Percentage  of  Deflection  due  to  IVeb  Members      =  41.4 

208.  The  Determination  of  Stresses  in  Redundant  Members.f — The  above  ready 
method  of  computing  deflections  accurately,  together  with  the  use  of  the  principle  expressed 
in  Proposition  III,  enables  us  to  find  the  stresses  in  redundant  members,  or  in  other  words  to 

*  The  member  Aa  being  a  heavy  tower,  it  is  not  included  in  the  table, 

t  For  a  very  complete  paper  by  Prof.  Wm.  Cain,  M.  Am.  Soc.  C.  E.,  on  this  method  of  finding  stresses  in  redun- 
dant members,  based  on  the  principle  of  least  work,  see  Trans.  Am.  Soc.  C.  E.,  Vol.  XXIV.,  p.  265  (Apr.  1891). 


228 


MODERN  FRAMED  STRUCTURES. 


solve  any  composite  system,  however  many  combinations  there  may  be  in  it.  To  do  this  we 
must  compute  the  deflections  of  each  elementary  system  for  a  given  load.  The  reciprocals  of 
those  deflections  represent  the  relative  rigidities  of  the  different  combinations,  and  since  the 
load  is  to  be  divided  in  proportion  to  these  reciprocals,  we  thus  obtain  one  less  number  of 
equations  than  we  have  systems.  The  other  equation  results  from  the  sum  of  the  parts 
equaling  the  whole.  This  then  gives  us  as  many  equations  as  there  are  systems,  and  we  can 
determine  what  part  of  the  total  load  passes  over  each  combination,  and  hence  solve  for  the 
stresses  in  such  combination.  If  any  one  member  forms  a  part  of  two  or  more  combinations, 
the  total  stress  in  it  is  the  sum  of  all  the  stresses  caused  by  the  several  combinations  of  which 
it  is  a  part.* 

The  application  of  the  formula        and  the  solution  for  redundant  members  will  be 

illustrated  by  an  example. 

By  putting  in  the  members  AG  and  BG,  Fig.  260,  the  system  becomes  composite.  The 

-4 


F       t?       G      aft  H 


Fig.  260.  Fig.  261. 


first  system  is  shown  in  Fig.  261,  and  the  second  in  Fig.  262.  The  members  AB  and  GD  are 
common  to  both  systems. 

The  lengths  of  the  members  are  given  on  the  left  half,  and  the  values  of  u  for  I  lb.  placed 
at  D  are  given  for  all  the  members  on  the  right  half  of  Figs.  261  and  262. 


Fig.  262.  Fig.  262a. 


The  load  at  D  will  divide  itself  between  the  systems  shown  in  Figs.  261  and  262;  the 
load  at  E  between  the  systems  261  and  262^,  and  similarly  with  the  load  at  C,  provided  CG 
and  GE  can  take  both  tensile  and  compressive  stress. 

By  Prop.  Ill  the  load  at  will  be  divided  between  the  two  systems  directly  as  their 
rigidities,  or  inversely  as  their  deflections  for  any  given  load.  But  when  the  joint  D  is  fully 
loaded  we  may  suppose  the  whole  bridge  is  fully  loaded.  In  this  case  all  the  members  would 
have  their  working  unit  stresses  which  may  be  supposed  to  be  the  same,  as  pi  for  all  the  ten- 
sion members  and  for  all  compression  members  in  the  combination.  At  least  the  parts 
should  be  proportioned  to  give  nearly  these  uniform  values,  and  it  will  be  here  assumed  that 
they  have  them. 

Since  we  wish  the  deflection  at  the  point  D,  we  put  the  i-lb.  load  there  for  finding  u,  the 
resulting  stress  in  each  member.  These  values  are  given  for  both  systems  on  the  right-hand 
portion  of  the  figures. 

*  In  the  case  of  initial  stress  in  counters  in  any  panel,  the  shear  in  this  panel  from  external  loads  divides  itself 
between  the  diagonals  by  increasing  the  stress  in  one  and  diminishing  it  in  the  other.  Thus  if  a\  and  oj  are  the  areas 
of  cross-section  of  the  two  counters,  and  .S"  is  the  shear  from  external  forces,  the  portion  of  this  shear  taken  by  is 

a\  Hi 

— ;  S  and  the  portion  taken  by  at  is  r — S,  being  additive  in  the  one  case  and  subtractive  in  the  other,  so  long  as 

oi  -)-  aa  <ii  -J-  ii 

both  remain  under  stress. 


DISTRIBUTION  OF  LOADS  OVER  REDUNDANT  MEMBERS.  229 


The  lengths  also  being  there  given,  and  pt  and     and  E  being  known,  we  may  write  at 

once  the  values  of       for  each  member : 

E 


Thus  for  the  first  truss  we  have 

For  lower  chord, 


pul  _  3^  pt 
E       h  '  E' 


For  upper  chord,        "   ~  ~h' 
For  verticals,  «   =  2^. 

 ' 

Whence  the  total  deflection  of  the  first  truss,  for  a  full  working  load,  producing  the  unit  stresses 
pt  and  p^  is 

Deflection  =  2^  =  ^^^'(A  (^4) 
E  hk 

For  the  second  truss,  for  a  full  working  load,  producing  the  unit  stresses  pt  and      we  have 

pul  _  \(P  pt 


For  the  lower  chord. 


h  E 


For  the  vertical,  "  —h. 

E 


For  inclined  struts. 
Whence  for  the  entire  truss 


h      '  £' 


Deflection  =  2^  =  ^^l^^l(p,  +  p^)  (15) 

These  two  deflections  are  equal,  as  seen  in  equations  (14)  and  (15),  when  -\-  2//'  =  4^i''  +  /i', 
or  when  d  =  h.    In  this  case  the  load  at  D  would  divide  itself  equally  between  the  systems. 

Next  taking  the  load  at  E,  and  the  two  systems  shown  in  Figs.  261  (load  at  E  instead  of 
at  D)  and  262a,  placing  now  the  i-lb.  load  at  E  in  each  system,  and  assuming  again  that  the 
distribution  of  this  load  is  desired  when  the  whole  bridge  is  loaded,  giving  a  Pt  unit  stress  in 
all  tension  members  and  a  unit  stress  in  all  compression  members,  it  can  be  shown,  as 
before,  that 

Deflection  at     of  first  system  (Fig.  261)       = — ^7— (/i+A)-    •    •    •  (^^) 

En 

Ad^  _j_ 

Deflection  at  E  of  second  system  (Fig.  262a)  =  ^ — •   •    •  i}7) 

U  d=  k,  these  become         4- p^)  and  ^^(p]  +  p,)  respectively. 

E  E 

Therefore  the  load  at  E  divides  itself  in  such  a  way  that  |  of  the  load  at  E  goes  on  the 
first  system  (Fig.  261)  and  |  on  the  second  system  (Fig.  262^). 

Evidently  the  same  would  hold  true  for  the  load  at  C.  In  other  words,  for  the  bridge 
fully  loaded,  and  for  d  =  h,  ^  the  load  at  D  and  f  of  the  loads  at  C  and  .fi"  would  be  carried 
by  the  Pratt  truss  system,  and  the  remainder  by  the  system  AGBEDC.  It  will  be  noted  also 
that  the  stress  in  GC  and  GE  is  compression  in  the  former  and  tension  in  the  latter,  and  hence 
the  real  stress  in  these  members  is  the  algebraic  sum  of  the  two. 


83° 


MODERN  FRAMED  STRUCTURES. 


From  the  above  it  is  evident  that  for  any  given  combination  of  members,  and  for  any 
given  system  of  loads,  it  can  be  determined  what  portion  of  the  total  load  goes  on  each 
system,  and  hence  what  the  stresses  are  in  every  member. 

209.  Direct  Measurement  of  Bridge  Strains. — To  learn  the  actual  strains  in  members 
of  a  bridge  truss,  or  the  strains  produced  by  rolling  loads  at  various  velocities,  some  kind  of 
apparatus  must  be  attached  directly  to  the  members  themselves,  and  their  elongation  or 
compression  noted  under  the  various  conditions  of  loading.    The  apparatus  shown  in  Fig.  263 


has  been  successfully  employed  in  Europe  on 
riveted  structures.  It  has  a  length  of  i  meter 
between  attachments,  and  registers  to  140  lbs. 
per  square  inch  (o.l  kg.  per  sq.  mm.)  when  read 
to  one  tenth  of  one  division  on  the  scale.  Any 
good  instrument-maker  could  readily  construct 
a  similar  one  to  give  stress  readings  by  means  of 
a  vernier  to  lOO  lbs.  per  square  inch.  If  the 
length  be  taken  as  five  feet,  this  would  require 
reading  this  length  to  the  nearest  0.0002  inch. 
The  diameter  of  the  upper  disk  should  be  about 
10  in.  and  the  length  of  indicator  about  6  in. 
In  working  out  the  relative  lengths  of  the  arms 
take  E  =  28,000,000.  The  method  of  attach- 
ment is  important.  It  should  be  fastened  by 
means  of  pointed  steel  set-screws.  For  eye- 
bars  this  is  an  easy  matter,  but  for  compression 
members  it  is  more  difficult.    The  attachment 


Fig.  263.  should  in  all  cases  be  symmetrically  made  on 

opposite  sides  of  the  member,  in  the  plane  of  its  neutral  axis.  It  should  be  graduated  to  read 
in  thousands  of  pounds  per  square  inch,  t 

209a.  Vibration  of  Bridges  from  Synchronous  Impacts. — Every  beam  or  truss 
carrying  a  given  total  load  has  a  definite  period  of  vibration  independent  of  its  amplitude. 
This  subject  has  been  investigated  theoretically  and  practically,*  and  the  periodic  time  of 
vibration  has  been  found  to  be 

t  =  0.0093/*  /y/ ~;  o    .    .  (16) 

where  t  =  time  of  vibration  in  seconds  ; 

/=  length  of  truss  or  beam  in  feet, 

/  =  total  load  in  pounds  on  one  truss  or  beam,  uniformly  distributed ; 

E  =  modulus  of  elasticity  of  the  material ; 

/  =  moment  of  inertia  of  the  truss  or  beam  in  foot-units. 
When  this  periodic  time  of  vibration  coincides  closely  with  any  synchronous  impact,  as 
the  stepping  of  a  horse,  or  the  revolution  of  unbalanced  locomotive  drivers,  or  the  passing  of  a 
low  joint  on  a  railway  bridge,  the  amplitude  of  the  vibration  rapidly  increases  until  a  com- 
\>^^^\.v^&\y  small  impact  may  by  repetition  produce  serious  deflections  with  their  corresponding 
stresses  in  the  members.  Diagrams  showing  this  action  on  railroad  bridges  have  been 
obtained  by  Prof.  S.  W.  Robinson  (Trans.  Am.  Soc.  C.  E.,  Vol.  XVI.,  p.  42),  who  has  also 
fully  discussed  the  problem. 


*  By  M.  Deslandres,  in  Annales  des  Fonts  et  ChaussSes  for  December,  1892.    His  equation  is  /'  =  6.5  — y,  where  /is 
_  I 

gravity,  <^  —  ^"d  kilogrammetre  units  are  used.  Various  diagrams  showing  the  effects  of  synchronous  impacts  are 
given. 

f  See  an  excellent  paper  on  this  subject,  giving  results  of  experiments,  in  Engineering  News,  May  g,  1895,  p.  300. 
These  results  show  that  the  actual  strains  agree  very  closely  with  the  theoretical  statically  computed  strains,  even  on 
the  hip-verticals,  under  a  speed  of  train  of  55  miles  per  hour. 


Part  II. 
STRUCTURAL  DESIGNING. 


Part  II. 


STRUCTURAL  DESIGNING. 


CHAPTER  XVI. 
STYLES  OF  STRUCTURES  AND  DETERMINING  CONDITIONS. 

210.  General  Considerations. — The  selection  of  the  proper  structure  to  use  to  fulfil 
given  conditions  is  a  problem  which  would  have  to  be  solved  as  a  special  case  for  any  general 
rules  which  may  be  established.  The  determining  factors  are  so  variable  that  experience  in 
the  location  and  selection  of  the  proper  structure  is  a  much  safer  guide  than  any  rigid  formulae. 
There  are,  however,  some  general  principles  and  approved  rules  which  are  worthy  of  attention 
and  which  may  be  used  without  any  very  great  error  by  those  who  lack  the  needed  experience. 
The  two  important  problems  which  confront  a  constructing  engineer  at  the  beginning  in  the 
building  of  a  new  bridge  are,  1st,  the  best  location  for  the  bridge;  and  2d,  the  proper 
structure  to  use.  The  items  of  first  cost  and  cost  of  maintenance  must  be  considered  together 
with  the  probable  life  of  the  bridge  and  its  safety  in  the  case  of  a  derailed  train.  In  general, 
it  may  be  assumed  that  the  item  of  first  cost  is  the  only  one  which  may  be  varied  for  any 
special  case  as  the  other  items  depend  upon  the  details  of  the  construction  which  would  be 
similar  for  various  locations  of  the  bridge  or  for  different  structures.  Economy  in  first  cost 
will  then  be  assumed  as  the  desired  result.  The  first  cost  of  a  bridge  will  vary  as  the  quanti- 
ties of  the  material  which  are  necessary  in  its  construction  vary.  The  quantities  which  may  be 
varied  are  the  masonry  in  the  piers  or  the  substructure,  and  ironwork  in  the  spans  or  the  super- 
structure. The  first  cost  will  be  a  minimum  when  the  combined  cost  of  the  substructure  and 
the  superstructure  is  a  minimum. 

211.  The  Selection  of  the  most  Economical  Location  for  a  Bridge. — It  will  be 
assumed  that  the  cost  of  building  the  road  to  any  of  the  crossings  under  consideration  is 
constant,  or  that  the  difference  in  cost  on  this  account  may  be  readily  estimated  and  taken 
into  account  in  the  comparisons.  When  there  are  no  local  conditions  limiting  the  length  of 
span  used  and  if  the  same  style  of  bridge  (i.e.,  deck  or  through)  may  be  used,  the  cheapest 
bridge  is  manifestly  the  shortest.  However,  if  the  spans  in  each  of  the  locations  were  fixed  by 
local  conditions  and  were  different,  the  shorter  bridge  requiring  the  longer  spans,  it  may 
be  possible  for  the  longer  bridge  to  be  the  cheaper.  The  amount  of  iron  in  a  single-track  pin- 
connected  span  is  very  closely  approximated  by  the  following  formula: 

W=  aP  +  kl, 

where  W  =  total  weight  of  iron  in  the  span,  /  =  length  of  the  span,  rt:  is  a  constant  which  may 
be  taken  as  5,  and  k  another  constant  which  varies  with  the  live  load  used  in  proportioning 
the  structure.  The  term  is  usually  assumed  to  represent  the  weight  of  the  trusses,  and 
kl  to  represent  the  weight  of  the  floor  system.  If  this  were  true,  both  a  and  k  would  vary 
with  the  live  load  used,  and  no  doubt  a  closer  approximation  to  the  weight  could  be  made  by 
varying  both  a  and  k,  but  in  an  approximate  formula  little  benefit  is  derived  from  such  refine, 
ment.  For  a  live  load  of  loO-ton  engines  with  the  usual  train  load  and  specifications  a  may 
be  taken  as  5  and  k  as  350. 

233 


234 


MODERN  FRAMED  STRUCTURES. 


The  weight  per  linear  foot,  w,  of  a  bridge  of  spans  of  /  length  would  be,  therefore, 

W 

w=~=  5/+  350. 

If  L  is  the  total  length  of  the  bridge,  the  total  iron  weight  would  be 

350). 

This  formula  could  be  used  to  find  the  iron  required  at  the  various  crossings,  and  there- 
fore the  cost  of  the  superstructure.  Combining  this  with  the  estimated  cost  of  the  substruc- 
ture, the  cheapest  crossing  can  be  very  closely  determined. 

As  an  illustration  of  the  application  of  this  formula  we  will  assume  alternative  single-track  crossmgs, 
one  of  which  requires  five  200-ft.  spans,  and  the  other  three  300-ft.  spans. 

For  the  five  200-ft.  spans,  w  =  1350,    Lw  =  1,350,000  lbs. 

For  the  three  300-ft.  spans  w  =  1850,    Lw  =  1,665,000  lbs. 

If  the  price  of  iron  was  five  cents  per  pound,  there  would  be  a  diflference  of  $15,750  in  favor  of  the 
longer  bridge  in  the  cost  of  the  iron.  If  this  were  more  than  enough  to  pay  for  the  two  extra  piers  required 
the  longer  bridge  would  be  the  cheaper, 

212,  The  Proper  Structure  to  Use  at  a  Given  Crossing. 

1st.  Bridges  of  Ofie  Span. — The  cost  will  vary  with  the  kind  of  bridge  used.  This  question 
will  be  considered  in  detail  in  a  following  article  on  the  kinds  of  bridges,  classifying  them 
according  to  their  method  of  construction  into  plate  girder  bridges,  riveted  truss  bridges,  and 
pin-connected  truss  bridges.  Assuming  the  kind  of  bridge  as  constant,  the  cost  will  depend 
upon  the  selection  of  the  cheaper  style  of  bridge,  i.e.,  deck  or  through.  The  deck  bridge  may 
generally  be  assumed  to  be  the  cheaper  style.    By  referring  to  Fig.  264,  it  will  be  seen  that 


Fig.  264. 

there  is  a  saving  in  the  height  of  the  necessary  piers  about  equal  to  the  depth  of  the  truss  due 
to  the  use  of  the  deck  span.  There  is  an  increase  in  the  weight  of  the  necessary  iron  of  about 
ten  per  cent  for  the  deck  span,  but  this  will  never  offset  the  saving  in  the  masonry.  Fig.  264 
represents  the  case  where  the  abutments  are  simple  piers,  the  approaches  to  the  span  being 
trestles  of  wood  or  iron.  Fig.  265  represents  the  case  where  the  abutments  are  retain- 
ing-walls.  There  would  generally  be  no  increase  necessary  in  the  masonry  on  account  of  the 
span  being  supported  on  the  retaining-walls,  so  that  the  choice  would  be  between  the  deck 
and  through  spans,  as  shown  in  the  figure.  There  would  be  very  liktle  saving,  if  any,  accom- 
plished by  using  the  deck  span  except  for  the  shorter  spans,  say  und^t  i&a£eet  long.  Fbf 


STYLES  OF  STRUCTURES  AND  DETERMINING  CONDITIONS.  235 


these  shorter  spans  the  trusses  of  the  deck  spans  may  be  placed  closer  together  than 
would  be  permissible  for  through  bridges  on  account  of  the  clearance  necessary  between  the 
trusses  of  the  latter.    This  would  reduce  the  length  of  the  floor-beams  and  save  some  iron. 


Fig.  265. 


The  iron  floor  system  may  be  dispensed  with  in  the  shorter  deck  spans  by  supporting  the 
cross-ties  directly  on  the  top  chord  and  a  further  reduction  in  the  required  amount  of  iron 
made. 

2d.  Bridges  which  may  be  of  One  or  More  Spans. — When  the  span  lengths  are  fixed  by  local 
conditions  the  problem  becomes  equivalent  to  that  of  a  series  of  bridges  of  one  span  each,  the 
only  open  question  being  the  kind  of  bridge  and  the  style  (i.e.,  deck  or  through).  When 
the  span  lengths  may  be  varied  and  the  style  the  same,  a  very  close  approximation  to  the 
correct  length  of  span  to  use  for  economy  in  total  cost  can  be  obtained  as  follows : 


Let  A 

cost  of  the  two  end  abutments  in  dollars  ; 

B 

—  cost  of  the  floor  and  that  part  of  the  iron  weight  which  remains  constant  in 

dollars  ; 

C 

—  cost  of  one  pier  in  dollars  assumed  as  constant ; 

I 

—  length  of  the  bridge  in  feet ; 

X 

—  number  of  spans  ; 

P 

—  price  of  iron  per  pound  in  dollars  ; 

y 

=  total  cost  of  bridge  in  dollars ; 

a 

—  weight  per  foot  of  a  span  b  feet  in  length. 

The  weight  represented  by  a  is  that  part  of  the  weight  per  linear  foot  which  varies  directly  as 
the  length  of  span  (see  Art.  211).    Then  the  weight  per  foot  of  a  bridge  with  spans  of  ^ 

length  will  be  7-  -,  and  the  total  cost  of  the  bridge  will  be 

y  =  A+B+{x-i)C+l^p. 
From  this  we  find^  to  be  a  minimum  when 


For  pin-connected  truss  spans  -  =  -.  If  iron  costs  five  cents  per  pound,  -  =  V4C;  and  if  iron 
costs  four  cents  per  pound,  -  =  V^C,    The  econoniical  lengths  of  spans,  in  feet,  (or  piers  of 


236 


MODERN  FRAMED  STRUCTURES. 


various  costs  are  given  in  the  following  table,  assuming  the  iron  to  cost  four  and  five  cents 
per  pound  respectively. 


Cost  of  One  Pier. 

Economical  Length  of  Span  in  Feet. 

Cost  of  One  Pier. 

Economical  Length  of  Span  in  Feet. 

Iron  4  cts.  per  lb. 

Iron  5  cts.  per  lb. 

Iron  4  cts.  per  lb. 

Iron  s  cts.  per  lb. 

$2000 

100 

89 

$10,000 

225 

200 

3000 

122 

no 

12,000 

245 

219 

4000 

141 

127 

14,000 

265 

237 

5000 

158 

141 

15,000 

283 

253 

6000 

173 

155 

1 8,  ODD 

300 

269 

7000 

187 

168 

20,000 

283 

8000 

200 

179 

24,000 

310 

As  the  length  of  the  bridge  is  fixed,  such  a  length  of  span  may  be  readily  selected  which 
will  make  the  total  cost  of  the  bridge  a  minimum. 

The  assumptions  made  in  deriving  the  formula  for  the  economical  length  of  span  are  not 
liable  to  be  in  error  enough  to  afTect  the  choice  of  the  proper  length  of  span  to  use  if  the  total 
length  of  the  bridge  is  fixed,  as  this  length  can  rarely  be  divided  into  an  even  number  of  spans 
of  the  economical  length.  It  is  well  to  err  by  choosing  a  longer  span  than  the  economical  one 
rather  than  a  shorter  span. 

213.  Kinds  of  Bridges. — The  metal  structures  in  common  use  in  the  railway  practice  of 
the  United  States  may  be  divided  into  the  following  kinds  according  to  their  method  of  con- 
struction : 

*  1st.  Plate  Girder  Bridges,  used  generally  up  to  75  feet  span  and  often  to  a  little  over 
100  feet. 

2d.  Riveted  Truss  or  Lattice  Bridges,  the  ordinary  lengths  of  which  range  between  70  and 
no  feet.  The  extremes  of  length,  however,  are  not  very  definitely  marked,  as  some  engineers 
use  them  as  short  as  50  feet  and  others  use  them  for  almost  any  length. 

3d.  Pin-connected  Truss  Bridges,  the  lengths  of  which  range  from  80  feet  to  the  longest 
span  in  use. 

A  review  of  the  generally  accepted  merits  of  each  of  these  kinds  of  construction  will  be 
given  for  the  purpose  of  aiding  the  student  or  engineer  in  the  selection,  when  occasion 
demands,  of  the  proper  structure.  Economy  will  be  the  deciding  factor  in  the  selection  of 
any  particular  design,  and  it  is  for  the  engineer  to  decide  what  is  true  economy  for  his  special 
case.  The  cost  of  the  maintenance  of  the  floor  and  the  stiffness  of  the  structure  against 
vibration,  which  is  a  measure  of  the  probable  life  of  the  bridge,  together  with  the  degree  of 
safety  insured  in  case  of  the  derailment  of  a  train,  are  items  which  affect  the  true  cost  in  addi- 
tion to  the  first  cost  and  should  have  due  consideration.  While  discussing  plate-girder 
construction  some  of  the  floors  in  general  use  will  be  described.  A  few  empirical  formulae  for 
the  iron  weights  of  bridges  will  also  be  given,  from  which  it  is  believed  that  it  will  be  safe  to 
derive  the  iron  weight  to  be  used  in  the  calculation  of  the  stresses,  and  also  which  will  show 
relatively  the  amount  of  iron  in  the  different  kinds.  These  formulae  have  been  found  to  give 
good  results  for  engine  loadings  of  about  one  hundred  tons  each  with  the  ordinary  train  load, 
rhey  also  assume  that  the  structures  are  carefully  designed  under  some  of  the  approved  general 
specifications  now  in  use,  and  that  the  bridge  is  not  askew,  that  the  alignment  of  track  on  the 
bridge  is  a  tangent,  and  that  the  design  is  the  most  economical  in  proportions  and  details. 

214.  The  Plate  Girder  Bridge  now  generally  used  for  spans  of  75  feet  or  less  is  deservedly 
the  most  popular  kind  of  construction  in  use.    There  is  a  minimum  opportunity  for  error  in 


*This  classification  to  include  rolled  I-beams  under  Plate  Girders. 


STYLES  OF  STRUCTURES  AND  DETERMINING  CONDITIONS  2^7 


its  design  and  calculation,  small  chance  for  defects  due  to  faulty  workmanship,  and  when  once 
put  in  place  requires  little  attention,  except  the  necessary  application  of  paint  to  prevent 
rusting.    It  is  the  simplest  and  cheapest  in  manufacture  for  short  spans,  but  for  lengths  over 


Fig.  266. 


60  feet  it  is  the  most  expensive  in  first  cost  of  all,  due  to  the  fact  that  it  requires  more 
iron  in  its  construction  than  any  other.  Its  use  is  gradually  extending  to  longer  spans  in 
spite  of  the  cost.    For  the  longer  spans  over  75  feet  the  girders  become  so  very  heavy  that 


Fig.  267. 


transportation  becomes  difificult.  The  longer  spans  require  deep  girders  for  economy  in  iron, 
and  the  necessary  web  plates  can  only  be  obtained  in  short  lengths,  making  frequent  web 
solices  necessary,  thus  adding  weight  and  increasing  cost.    It  is  always  preferable  to  have  plate 


Fig.  268. 


MODERN  FRAMED  STRUCTURES. 


girders  riveted  up  completely  in  the  shop  by  power,  leaving  the  bracing  between  the  girders  as 
the  only  parts  requiring  hand  riveting  after  the  girders  are  in  place  on  the  piers. 

Fig.  266  shows  cross-sections  at  ends  and  in  middle  of  span  for  a  single-track  deck  plate 
girder  bridge  with  the  usual  floor  and  mode  of  attachment.  The  floor  shown  is  the  one  most 
generally  used.    The  detail  construction  of  the  ironwork  is  simple. 

Fig  267  shows  a  cross-section  of  the  most  common  style  of  plate  girder  through  bridge 
ivith  the  same  floor  and  method  of  attachment  as  was  shown  in  Fig.  266. 

Fig.  268  shows  the  cross-section  of  two  styles  of  through  plate  girder  bridge  in  which  the  iron 
floor  system  shown  in  Fig.  267  is  dispensed  with  and  the  track  supported  on  large  cross-ties  rest- 
ing on  shelf-angles  riveted  to  the  webs  of  the  girders  or  on  the  bottom  flange  angles. 

Fig.  269  shows  longitudi  al  sectional  views  of  through  plate  girders  with  various  kinds  of 
the  solid  iron  floor  on  which  a  ballasted  roadbed  is  carried  over  the  bridge. 


Fig.  26g. 


The  iron  weights  of  the  various  styles  of  plate  girder  bridges  *  may  be  approximated  by 
the  following  formulae: 

Deck  Plate  Girder  Bridge,  w  =  gl  -\-  \  \o\ 

Through  Plate  Girder  Bridge,  w  —  8f  /  -|-  300  (iron  floor  system) ; 

"  "        "         "  w  =  g\l  -\-  150  (large  ties  on  shelf  or  flange  angles); 

"  "         "  w  =  10/  -|-  600  (solid  iron  floor) ; 

when  zv  =  iron  weight  per  linear  foot,  /  =  length  over  all  of  girders  (i.e.,  length  of  necessary 
bed  plates  plus  distance  centre  to  centre  of  bed  plates). 

215.  Bridge  Floors. — Figs.  266,  267,  268,  and  269  show  nearly  all  the  floors  now  much 
used.  Attention  will  be  called,  however,  to  a  very  popular  modification  of  the  floor  shown  in 
Fig.  267,  which  consists  of  the  addition  of  safety  stringers  or  two  additional  stringers  placed 
from  2  to  4  feet  outside  of  the  main  stringers.  They  are  usually  made  one  half  the  strength 
of  the  main  stringers,  and  arc  used  for  safety  in  case  of  the  derailment  of  the  train.  Some- 
times the  main  trusses  or  girders  of  the  bridge  are  used  in  place  of  the  safety  stringers,  as,  for 
example,  in  Fig.  267  the  cross-ties  may  be  extended  to  rest  on  shelf-angles  riveted  to  the  webs 
of  the  girders. 

The  floor  shown  in  Fig.  268  may  be  called  the  standard  and  is  the  most  generally  used  of 
all.  The  details  of  the  guard-rails  and  the  many  various  methods  of  attaching  cross-ties  and 
guards  to  the  supporting  ironwork  will  not  be  considered.  The  usual  attachment  is  to  fasten 
ties  and  guards  to  the  stringer  by  means  of  a  three-quarter-inch  hook-bolt  passing  through 
every  third  or  fourth  cross-tie.  The  wooden  guard-rails  are  spiked  or  bolted  to  every  tie. 
The  wooden  guard  rail  is  usually  placed  with  its  inside  face  about  I  foot  from  inside  face  of  rail 
or  gauge.  The  inner  upper  corner  is  sometimes  covered  by  an  angle-iron  put  on  with  countersunk- 
head  spikes.    Cross-ties  are  usually  spaced  with  openings  of  from  4  to  6  inches  between  them. 


*  Single  track. 


STYLES  OF  STRUCTURES  AND  DETERMINING  CONDITIONS.  239 


The  floors  shown  in  Fig.  268  arc  used  when  a  cheap  bridge  is  wanted  and  when  the 
distance  from  the  rail  to  the  underside  of  the  bridge  is  Hmited.  The  objection  to  them  is  that 
the  cross-tie  is  under  strain  from  the  load  and  must  necessarily  be  very  large  and  subject  to 
continual  inspection,  and  be  promptly  removed  in  case  any  defect  which  impairs  its  strength 
is  noticed.  The  replacement  of  the  cross-ties  is  a  difficult  task,  and  in  order  that  they  may  be 
put  in  at  all  the  stiffener  angles  on  the  inside  of  the  web  plate  must  be  so  spaced  as  to  allow, 
room  to  get  them  in.  The  size  of  cross-tie  to  use  is  determined  by  the  load  to  be  carried 
assuming  the  heaviest  axle  load  to  be  supported  by  three  ties,  allowing  an  extreme  fibre 
stress  of  from  800  to  1000  lbs.  per  square  inch. 

The  solid  iron  floors  shown  in  Fig.  269  are  not  very  generally  used  as  yet,  the  railroads 
which  do  use  them  much  being  those  in  the  North  and  East.  They  are  expensive  owing  to  the 
amount  of  iron  necessary  in  their  construction,  but  are  undoubtedly  the  safest,  most  rigid  and 
permanent  of  all  the  various  kinds.  They  can  be  built  when  the  distance  from  the  rail  to  the 
lowest  part  of  the  bridge  is  very  small.  With  this  kind  of  floor  the  corrugations  are  filled  with 
concrete,  usually  bituminous,  and  the  standard  ballast  on  top  of  this.  The  cross-tie  may  be 
embedded  in  concrete  in  the  corrugations  in  case  the  floor  must  be  very  shallow.  The  iron  in 
corrugated  floors  alone  weighs  from  thirty  to  forty  pounds  per  square  foot  of  floor. 

216.  Riveted  Truss  or  Lattice  Bridges  are  used  for  the  shorter  spans  on  account  of 
their  cheapness,  the  plate  girder  being  the  only  allowable  substitute,  while  for  the  longer 
spans  when  they  sometimes  supplant  pin-connected  trusses  it  is  generally  so  because  of  a 
prevailing  belief  that  they  are  stiffer  structures  and  safer  under  a  derailed  train.  There  is  a 
range  between  the  limits  of  about  80  and  100  feet,  in  which  the  riveted  truss  is  undoubtedly 
preferable  to  the  pin-connected,  but  beyond  that  it  is  very  doubtful  whetiier  it  is  correct  to 
adopt  it  in  preference  to  a  well-designed  pin-connected  bridge  if  the  price  is  in  the  latter's 
favor,  as  it  usually  will  be.  Either  kind  of  construction  will  make  satisfactory  bridges  for  the 
longer  spans  if  well  designed. 

In  the  design  of  riveted  trusses  the  use  of  multiple  systems  of  web  bracing  has  become 
generally  accepted  good  practice,  and  now  it  is  the  preferable  bracing  for  all  lengths  of  span. 
In  the  multiple  intersection  trusses  the  effect  of  the  distortion  of  the  truss  under  load  at  the 
joints  of  the  web  and  chord  members  is  much  reduced  by  the  rigid  joints  made  at  the  inter- 
sections of  the  web  members  themselves.  For  very  deep  trusses  the  effect  of  the  distortion  is 
also  small,  so  that  single  intersection  trusses  will  make  good  bridges. 

The  secondary  stresses  in  the  riveted  truss  make  the  calculation  and  design  of  them  very 
much  more  unsatisfactory  than  for  a  pin-connected  truss. 

The  riveted  truss  may  be  built  with  T-chords  or  box-chords  (Fig.  270),  the  latter  being 
the  more  expensive,  but  generally  the  preferable  design. 


Box-chords.  Fig.  270.  T-chords. 


For  "  pony  trusses"  or  half-through  bridges  the  box-chord  will  always  make  the  better 
bridge.    The  web  bracing  should  always  be  symmetrical  with  the  plane  of  the  centre  of  the 


240 


MODERN  FRAMED  STRUCTURES. 


truss,  and  care  should  be  taken  to  get  the  neutral  axes  of  all  members  meeting  at  a  joint  to 
intersect  at  one  point. 

The  riveted  truss  requires  more  metal  in  its  construction  than  the  pin-connected,  and  its 
manufacture  is  nearly  as  expensive.  For  long  spans  the  trusses  of  which  cannot  be  shipped 
riveted  up  complete,  all  connections  of  web  members  with  the  chords  must  be  hand-riveted. 

The  iron  weights  of  riveted  truss  bridges*  may  be  approximated  by  the  following  formulae 

Deck  bridge,  cross-ties  on  top  chord   w  =  200; 

Through  bridge,  iron  floor  system   w  =  7/  -|-  300; 

w  =  iron  weight  per  linear  foot ;  /  =  length  centre  to  centre  of  bearings. 

217.  Pin-connected  Truss  Bridges  are  more  generally  used  than  the  riveted,  and  owe 

their  popularity  to  their  cheapness,  facility  of  erection,  and  adaptability  to  almost  all  con- 
ditions. They  are  used  for  all  spans  over  80  feet,  although  the  general  practice  now  is  to 
limit  them  to  spans  over  100  feet.  In  the  early  days  of  bridge  building  the  use  of  the  pin 
connection  was  imperative,  but  of  late  years  the  facilities  of  constructors  in  general  for  handling 
larger  pieces  and  for  doing  better  field  work  in  erecting  and  riveting,  and  the  capacity  of  the 
railroads  for  transporting  heavier  and  longer  pieces  than  formerly,  have  made  it  possible  to 
build  better  riveted  trusses.  It  is  chiefly  because  of  this  that  the  riveted  truss  is  supplanting 
the  pin-connected  for  the  shorter  spans  where  short  panels  and  very  light  trusses  would  make 
the  latter  expensive.  The  pin-connected  truss  can  be  more  satisfactorily  designed  and  pro- 
portioned because  of  its  almost  absolute  freedom  from  secondary  stresses. 

The  single  intersection  truss  is  now  the  accepted  practice  in  the  construction  of  pin- 
connected  truss  spans,  the  recent  use  of  the  inclined  top  chord  correctly  and  the  Petit  truss 
making  it  possible  to  build  long  spans  cheaply  owing  to  the  lighter  and  shorter  members  that 
the  latter  types  require. 

The  two  general  types  of  truss  in  common  use  are  the  Warren  or  Triangular  truss  and 
the  Pratt  truss.  Another  type  of  truss  deserving  of  mention  is  the  "  Pegram  truss,  designed 
and  patented  by  Geo.  H.  Pegram,  M.A.S.C.E.    It  is  fully  described  in  Part  I  of  this  book. 

218.  The  Warren  or  Triangular  Truss  usually  requires  less  material  in  its  construction 
than  the  Pratt,  and  for  short  spans,  particularly  deck  spans,  is  commonly  used.  It  is  also  often 
employed  when  it  is  desirable  to  avoid  adjustable  members  in  the  truss,  because  the  symmetry 
of  the  truss  is  maintained  and  a  much  better  looking  structure  secured  than  if  the  Pratt  were 
used  with  stiffened  ties  in  place  of  counter  ties.  An  objection  often  urged  against  the  Warren 
is  that  the  continual  reversal  of  stress  in  the  web  members  causes  the  bearings  on  the  pins  to 
wear  or  indent  the  latter.  An  increase  of  the  bearing  surface  on  the  pins  would  be  the  remedy 
for  this.  A  great  objection  to  this  truss  for  through  spans  is  that  the  floor-beams  must  either 
be  suspended  from  the  pins  or  vertical  posts  introduced  at  each  panel  point  to  rivet  the  beams 
to  above  the  pin  ;  the  former  requirement  making  the  design  of  an  efificient  lateral  system 
difificult,  and  the  later  adding  material  with  very  little  compensation.  This  truss  is  now  rarely 
used  for  long  spans. 

219.  The  Pratt  Truss,  including  in  this  type  all  single  intersection  trusses  with  vertical 
intermediate  posts,  is  by  far  the  truss  most  generally  used.  It  may  be  termed  the  standard 
truss  in  American  practice.  In  its  details  it  is  simpler  than  any  other  form,  and  has  stood  the 
test  of  continual  use  without  any  diminution  of  its  popularity.  In  economy  of  material  and 
cost  it  is  only  second  to  the  Warren  for  the  shorter  spans,  while  for  the  longer  inclined  chord 
and  Petit  truss  spans  it  requires  less  material  and  is  less  expensive.  The  recent  long  spaas 
built  in  this  country  are  generally  Petit  trusses. 


*  Single  track. 


STYLES  OF  STRUCTURES  AND  DETERMINING  CONDITIONS. 


The  iron  weight  in  pin-connected  bridges*  may  be  approximated  by  the  following  for- 
mulae: 


220.  Swing  Bridges- — For  clear  openings  under  60  feet  the  plate  girder  bridge  is 
generally  used,  the  alternative  being  the  riveted  truss  for  these  shorter  spans.  For  more  than 
60  feet  clear  opening  the  pin-connected  truss  is  used  because  of  its  cheapness,  and  as  it  also 
has  necessarily  the  advantages  of  stifif  top  and  bottom  chords. 

221.  Cantilever  Bridges. — Cantilever  bridges  are  now  used  only  in  those  cases  where  it 
is  impossible  or  impracticable  to  erect  simple  spans.  They  are  not  economical  in  material,  as 
they  rarely,  if  ever,  require  as  little  iron  in  their  construction  as  a  simple  span.  Owing  to 
their  lack  of  rigidity,  a  simple  span  bridge  is  preferred  if  it  is  practicable  to  build  one  at  the 
crossing. 

222.  The  Three-hinged  Arch  Bridge  will  very  probably  be  preferred  in  those  cases 
where  the  bridge  must  be  erected  without  any  temporary  false-work,  and  where  the  abutments 
may  easily  be  made  sufificient  to  take  the  thrust.  A  bridge  of  this  kind  can  readily  be 
designed  to  erect  as  a  cantilever.  The  three-hinged  arch  requires  less  material  in  its  construc- 
tion than  a  simple  span. 

223.  For  Long  Spans,  including  those  over  600  feet,  the  cantilever,  the  arch,  the  canti- 
lever-arch, and  cantilever-suspension  are  the  types  from  which  the  engineer  has  now  to  select 
the  proper  structure.  The  cantilever-arch  and  cantilever-suspension  bridges  require  the  least 
quantity  of  material,  and  are  probably  the  designs  to  be  depended  upon  for  spans  longer  than 
the  practical  limit  set  for  simple  spans. 


Deck  span,  cross-ties  on  top  chord 

"       "    iron  floor  system  

Through  span,  iron  floor  system . . 


w=  5/-f25o; 
w  =  5.0/  +  475  ; 
■w  =  5.0/+  350. 


*  Single  track. 


•42  MODERN  FRAMED  STRUCTURES. 


CHAPTER  XVII. 


DESIGN  OF  INDIVIDUAL  TRUSS  MEMBERS. 


224.  The  Fatigue  of  Metals.* — Elaborate  experiments  in  Germany  and  elsewhere 
have  shown  that  the  ultimate  strength  of  metal  from  a  single  test  is  no  indication  of  its  ability 
to  resist  repeated  stresses. 

If  y  =  initial  unit  strength  ; 
A  —  greatest  unit  load  that  may  be  repeated  an  unlimited  number  of  times  ; 
Pi  —  greatest  unit  stress  that  can  be  reversed  an  unlimited  number  of  times  ; 
then  we  may  call 

/  the  initial  strength  ; 
/>,  the  repetition  limit ; 
/,  the  reversal  limit. 

That  is  to  say,  is  the  greatest  unit  stress  that  can  be  wholly  removed  an  unlimited 
number  of  times,  while  is  the  greatest  unit  stress  which  can  alternate  from  tension  to 
compression  an  unlimited  number  of  times.  Both  />,  and  />,  are  below  the  elastic  limits  for 
wrought-iron, /,  being  about  26,000  pounds  per  square  inch  and  about  16,000  pounds  per 
square  inch.  The  ultimate  strength  may  be  put  at  50,000  pounds  and  the  elastic  limit  at 
30,000  pounds  per  square  inch. 

But  in  practice  the  maximum  load  is  seldom  wholly  removed,  and  often  the  reversed 
stresses  are  not  equal,  so  that  in  general  we  may  say  that  we  have  a  maximum  unit  stress  m 
and  a  minimum  unit  stress  tt.  This  minimum  stress  may  or  may  not  be  of  the  same  kind 
(tension  or  compression)  as  the  maximum  stress.  For  these  general  cases  it  is  desirable  to 
know  what  the  values  of     and  n  may  be  for  any  material. 

In  Fig.  271  the  relation  of  these  limiting  and  working  stresses  is  shown 
graphically.  Distance  along  the  line  represents  stress  per  square  inch, 
measured  either  way  from  0  as  an  origin. 

Thus  :       of  =  f  —  initial  strength  ; 

op^  =  p^  =  repetition  limit ; 
op^  —  p^—  reversal  limit. 

The  stress  of  represents  the  strength  of  the  material  from  a  single  test. 
The  point  {e.l.)  is  the  primary  elastic  limit,  but  it  plays  no  important  part 
in  this  discussion.  The  stress  op^  may  be  put  wholly  on  and  off  an  unlimited 
number  of  times,  and  the  stress  op  ^  maybe  changed  to  op^'  an  unlimited 
number  of  times. 

If  all  the  stress  is  not  removed  each  time,  but  only  a  part  of  it,  then 
the  maximum  stress  7n  may  be  greater  than  />, ,  so  that  if  a  certain  portion 
of  oin,  represented  by  on,  be  left  on  permanently,  then  in  will  lie  somewhere 
in  the  field  />,/ ;  and  the  greater  is  the  ratio  of  the  fixed  to  the  varying 
stress,  the  more  nearly  will  in  approach  /.  Similarly,  if  only  a  part  of  the 
stress  is  repeated  with  the  opposite  sign,  then  the  greater  of  the  two 
stresses,  lie  somewhere'^n"  the  field  /.^^ ,  and  the  less,     ,  will  lie  between  0  and  //  , 


(eJ:)- 


/  \ 


/  ; 

o-'n'"  I 


/ 


Fig.  271. 


*  This  and  the  following  ariicle  are  from  a  paper  by  Prof.  Johnson,  read  before  the  Engineers'  Club  of  St.  Louis, 
November  16,  1SS7.  For  a  much  ful.er  discussion  of  this  question  ste  Chap.  XXVII  of  Prof.  Johnson's  Malei  ials  of 
Conslriulion  (John  Wiley  &  Sons,  1897). 


DESIGN  OF  INDIVIDUAL  TRUSS  MEMBERS. 


243 


and  the  more  nearly  n  is  numerically  equal  to  m,  the  more  nearly  will  m  approach  to 
Thus  we  may  say  that  the  maximum  stress,  m,  will  always  be     plus  a  portion  of  /j/when  n 
is  of  the  same  kind  of  stress,  and  minus  a  portion  of        when  n  is  of  the  opposite  kind  of 
stress. 

It  has  been  found  by  experiment  that  the  following  is  approximately  true :  The  maximum 
stress  is  equal  to  the  repetition  stress  plus  or  mimes  such  a  part  of  the  adjacent  field  as  the 
minimum  stress  is  a  part  of  the  maximum  stress.  Or, 

n 

m  =  p^-{-  -^f  —  p^  for  repeated  stresses,  (i) 

and 

m  =  pi  —         —  A)  for  reversed  stresses  (2) 

These  are  the  formulae  to  use  for  determining  the  breaking  stress  m  when  the  smaller  and 
fixed  stress  n  is  known  and  when  these  stresses  succeed  each  other  an  unlimited  number  of 
times. 

This  is  also  shown  in  Fig.  271. 

Thus  when  n  lies  above  0,  m  is  above     ;       when  n  lies  below  0,  m  is  below  /,  ; 

"     71  is  at  0,  m  is  at  /, ;  "     n  is  at      (=  —  /,),  m  is  at  />, ; 

"     n  is  at  m  (static  load),  m  is  at  f. 

Evidently  m  —  71  is  always  the  portion  of  the  stress  removed  each  time,  corresponding  to 
the  movable  load  on  bridges. 

For  every  variation  of  stress  there  is  a  corresponding  distortion,  and  the  product  of  the 
mean  value  of  the  variable  stress  into  the  distortion  is  the  work,  in  foot-pounds,  done  on  the 
material  in  distorting  it.  When  the  stress  is  partly  or  wholly  removed  the  member  recovers  a 
corresponding  portion  of  its  distortion,  and  this  is  work  done  by  the  member  against  the 
external  forces.  Now  it  is  this  zvork  which  ivcars  out  or  fatigites  the  material.  A  given 
material  can  recover  its  length  an  infinite  number  of  times  if  the  work  demanded  each  time 
be  not  too  great,  and  hence  it  is  capable  of  doing  an  infinite  amount  of  work  if  done  in 
sufificiently  small  amounts.  If  too  much  be  required  at  any  one  time,  however,  then  it  wears 
out  or  becomes  fatigued,  and  finally  breaks  down,  very  much  the  same  as  an  overworked 
muscle. 

Now  the  amount  of  work  done  at  any  one  time  has  been  shown  to  be  the  mean  stress 
into  the  distortion.  In  order  to  keep  this  maximum  single  effort  a  constant,  it  is  evident  that, 
as  the  mean  stress  increases,  the  distortion  must  diminish.  In  other  words,  as  the  maximum 
load  increases,  the  variation  in  load  (;«  —  «)  must  decrease.  But  for  711  increasing,  i>t  —  71 
can  decrease  only  by  the  more  rapid  increase  in  71  ;  therefore,  it  is  only  by  increasing  the 
static  load  n  that  the  total  load  ;«  maybe  raised  above And  since  a  single  effort,  equal  to 
/,  will  rupture  the  piece,  it  is  evident  that  as  7n  approaches /  the  limits  between  which  we  can 
continue  to  work  our  specimen  indefinitely  will  become  narrower  by  the  approach  of  n 
towards 

Similarly,  as  the  lower  limit  passes  below  the  zero  point  and  is  therefore  changed  into  a 
stress  of  the  opposite  sign,  the  mean  value  of  the  stress  diminishes,  and  hence  the  distance 
through  which  the  piece  can  be  worked  increases,  this  maximum  range  being  2/,  when  n  —  p^ 
and  w  =  p^. 

These  "new  formulae"  for  dimensioning  are  therefore  seen  to  be  very  simple  in  form  and 
rational  in  conception. 

225.  Working  Formulae. — Formula  (i)  (named  after  Prof.  Launhardt)  may  be  put  in 
the  form 

J.  I     I  stress  \ 

=  A  '  +  ^   ;  1  (3) 

\  /,       max.  stress  / 


244  MODERN  FRAMED  STRUCTURES. 

and  formula  (2)  (named  after  Prof.  Weyrauch)  may  be  put  in  the  form 


m  =  p\\ 


/■  -A 


min.  stressX 
max.  stress/' 


(4) 


Experiments  made  by  Wohler,  Spangenberg,  Bauschinger,  and  Baker  show  that  for 
structural  iron  and  steel  is  approximately  ^f,  and  is  approximately  \f^\p,. 
Therefore  we  have 


For  stresses  of  one  kind  m 


I       min.  stress\ 
^■V+max.stressJ' <5) 


and 


For  stress  of  opposite  kinds  m 


=4 


1  min.  stressX 

2  max.  stress/ 


(6) 


110000 


Since/,  is  approximately  one  half  the  ultimate  strength  for  wrought-iron  and  mild  steel, 

and  is  a  common  factor  in  the  right-hand  members 
of  these  equations,  the  factor  of  safety  may  be  in- 
troduced in  />, ,  as,  for  instance,  for  like  stresses,  the 
r        I     .  (     ,  min.N 

formula  /  =  9000(^1  + -^^)  would  imply  a  factor 

of  safety  of  three  for  an  indefinite  number  of  repeti- 
tions of  the  maximum  load  in  wrought-iron. 


100000 


90000^ 


Fig.  272. 


226.  The  Usual  Methods  of  Proportioning  Individual  Truss  Members  Subjected 
to  Varying  Direct  Stresses. — It  is  only  within  recent  years  that  the  fatigue  of  the  metal 
due  to  varying  stress  has  been  taken  into  consideration  in  proportioning  individual  truss 
members  in  which  the  stress  is  always  of  one  kind,  either  tension  or  compression.  It  is  not 
the  universal  practice  at  present.  If  the  stress  varied  in  kind,  the  practice  has  been  to  use  a 
lower  stress  per  square  inch  on  such  members. 


DESIGN  OF  INDIVIDUAL  TRUSS  MEMBERS. 


245 


The  more  general  practice  is  to  proportion  compression  members  without  any  reference 
to  the  fatigue  of  the  metal,  and  in  many  specifications  tension  members  are  treated  in  the 
same  way.  The  Pennsylvania  Railroad  Company  was  probably  the  first  to  introduce  in  its 
specifications  formulae  which  were  based  on  the  fatigue  of  the  metal  from  varying  stress.  For 
varying  stresses  of  the  same  kind,  all  compression  or  all  tension,  the  company's  formula  is 


minimum  stress' 
maximum  stress^ 


when  b  =  allowed  stress  per  square  inch  in  pounds,  and  «  is  a  constant  determined  by  the 
quality  of  the  material  used  and  the  kind  of  stress,  tension  or  compression,  being  7500  for 
double  rolled  iron  in  tension,  7000  for  plates  or  shapes  in  tension,  and  6500  for  plates  or 
shapes  in  compression.    For  varying  stresses  of  opposite  kinds, 

,        /  maximum  stress  of  lesser  kind  \ 

b  =  a\i  —  :  — -,  . 

\       2  .  maximum  stress  of  greater  kind/ 

For  compression  b  is  to  be  still  further  reduced  by  Gordon's  formula;  for  long  struts. 

Cooper's  specifications  allow  a  unit  stress  twice  as  great  for  dead  load  as  for  live  load,  and 
thus  give  practically  the  same  results  as  those  obtained  by  the  Pennsylvania  R.  R.  formulae. 
For  varying  stresses  of  opposite  kinds  Cooper  specifies  that  the  maximum  stress  of  either 
kind  shall  be  increased  by  eight  tenths  of  the  maximum  stress  of  the  lesser  kind  and  the  piece 
proportioned  by  the  usual  formula;  for  whichever  of  these  combined  stresses  would  require 
the  larger  area  of  cross-section.  Thus,  if  a  piece  is  subjected  to  alternating  stresses  of 
100,000  pounds  tension  and  80,000  pounds  compression,  it  must  be  proportioned  for  164,000 
pounds  tension  or  144,000  pounds  compression,  whichever  requires  the  larger  area  of  cross- 
section. 

The  old  method  of  proportioning  a  piece  subjected  to  alternating  stresses  is  to  increase 
the  maximum  stress  of  cither  kind  by  eight  tenths  of  the  maximum  stress  of  the  other  kind, 
and  to  proportion  the  piece  by  the  usual  formula;  for  whichever  of  these  combined  stresses 
requires  the  larger  area  of  cross-section.  It  is  evident  that  numerical  formulae  in  the  form  of 
(5)  and  (6)  arc  the  most  rational  that  can  be  used. 

227.  Tension  Members  may  be  divided  into  four  kinds,  according  to  their  method  of 
construction  : 

1st.  Eyebars. 

2d.  Square  or  round  rods. 

3d.  Single  shapes. 

4th.  Compounded  sections. 

1st.  Eyebars  are  used  for  the  main  tension  members  of  pin-connected  trusses.  They 
are  now  generally  made  of  low  steel  of  an  ultimate  strength  of  from  56,000  to  66,000  pounds 
per  square  inch,  the  methods  of  manufacture  securing  a  more  satisfactory  and  reliable  product 
from  that  metal  than  from  iron.  Steel  eyebars  are  made  by  forging  or  upsetting  the  eye  or 
head  of  the  bar  in  a  die,  and  subsequently  reheating  and  annealing  the  finished  bars  previous 
to  boring  the  pin-holes.  The  quality  of  the  finished  bar  depends  principally  on  the  annealing 
process.  Wrougiit-iron  cj  ebars  are  made  by  piling  and  welding  the  head  or  eye  on  the  body 
of  the  bar  by  various  methods.  Welding  is  always  an  unreliable  process,  and  defects  due  to 
imperfect  welds  and  burnt  material  are  difficult  to  detect. 

In  the  following  table  are  given  the  standard  sizes  of  steel  eyebars  (i.e.,  width  of  bar  and 
diameter  of  eye)  manufactured  by  the  Edge  Moor  Bridge  Works  and  which  fairly  represent 
the  present  practice. 


246 


MODERN  FRAMED  STRUCTURES. 


It  will  be  noted  that  there  is  a  specified  limiting  minimum  thickness  of  bar.  This  has 
been  found  to  be  advisable  on  account  of  the  manufacture,  and  because  thin  bars  will  buckle 
in  the  head  when  under  strain.  This  minimum  thickness  increases  with  the  width  of  the  bar 
and  the  diameter  of  the  eye.  The  thickness  of  the  bar  may  be  made  anything  greater  than 
this  minimum,  but  a  thickness  of  two  inches  for  bars  six  inches  wide  and  under  is  rarely 
exceeded.  As  it  is  desirable  to  use  small  pins,  and  therefore  small  eyes,  on  account  of  the 
cost  of  manufacture  and  the  amount  of  material  needed  to  make  the  eyes,  a  thickness  of  bar 
will  generally  be  selected  that  will  make  these  results  possible.  The  thicker  the  bar  the  greater 
will  be  its  leverage  and  stress,  and  consequently  the  greater  will  be  the  bending  moment  on 
the  pin  which  would  result  in  requiring  a  larger  pin  and  a  larger  eye. 


EDGE  MOOR  STANDARD  STEEL  EYEBARS. 
11 


A 

B 

E 

D 

C 

A 

B 

E 

D 

C 

Width  of  Body 
of  Bar. 

Minimum 
Thickness  of 
Bar. 

Diameter  of 
Head  or 
Eye 

Diameter  of 
Largest  Pin 
hole. 

See  Note. 

Width  of 
Body  of  Bar. 

Minimum 
Thickness  of 
Bar. 

Diameter  of 
Head  or 
Eye. 

Diameter  of 
Largest  Pin- 
hole. 

See  Note. 

3 
3 

3 
4 

4 
4 
5 
5 
5 
5 

i 
i 
f 
i 
i 
I 
i 

i 

I 
I 

b\ 

8 
9 

9i 

\o\ 

Hi 

13 
14 

2i 

4 

5 

5i 
64 
4f 
5f 
6J 
1\ 

33 
33 
33 
33 
33 
33 
37 
37 
37 
37 

6 
6 
6 

7 
7 
8 
8 
8 
9 
9 

7 

1!" 
7 

I 

1  5 

TS 

\\ 

I 

I 

I 

14 

li 

I3i 

Mi 

I5i 

•5i 

17 

17 

18 

19 
i9i 

2li 

5i 

6i 

7i 

54 

74 

5f 

6f 

8 

7 

9 

37 
37 
37 
40 
40 
40 
40 
37 
39 
39 

Note. — In  column  C  are  given  the  percentages  of  excess  of  area  of  the  eye  on  line  SS  over  the  area  of  the  body  of 
the  bar  when  the  largest  pin-hole  is  in  the  eye. 


It  is  always  better  to  use  an  eye  the  diameter  of  which  is  about  two  and  one  quarter 
times  the  width  of  the  bar.  In  extreme  cases  the  diatneter  of  the  eye  may  be  made  two  and 
one  half  tiines  the  width  of  the  bar,  but  it  is  never  desirable  to  exceed  this,  as  the  cost  and 
difficulty  of  manufacture  increase  rapidly  if  larger  eyes  are  used. 

The  amount  of  material  required  to  make  an  eye,  or  the  extra  length  of  bar  required 

beyond  the  centre  of  the  pin,  is  approximately  L  —  ,  when  L  =  length  of  bar  required 

beyond  the  centre  of  the  pin  and  E  and  A  as  given  in  the  table. 

Eyebars  are  now  made  as  large  as  12  X  3  inches  with  eyes  27  to  30  inches  in  diameter. 

The  tests  of  full-sized  eyebars  do  not,  as  a  rule,  give  as  high  results  in  ultimate  strength  as  the  small 
specimen  tests  of  the  material.  The  small  specimen  test  pieces  have  usually  an  area  of  one  half  of  one 
square  inch.  This  difference  in  the  results  of  the  tests  has  caused  a  great  amount  of  friction  between 
manufacturer  and  customer.  Recent  study  of  the  subject  shows  that  a  difference  in  the  ultimate  strength  of 


DESIGN  OF  INDIVIDUAL  TRUSS  MEMBERS. 


247 


the  small  test  piece  and  the  full  size  bar  is  to  be  expected.  Mr.  F.  H.  Lewis,  M.A.S.C.E.*,  after  an  ex- 
haustive study  of  this  problem,  finds  that  the  losses  in  ultimate  strength  of  full  sized  eyebars  are  due  to  three 
distinct  causes,  viz.: 

1.  The  small  specimens  are  so  cut  from  the  original  bar  as  to  give  results  which  are  in  excess  of  the 
average  value  of  the  bar. 

2.  There  is  a  legitimate  loss  in  ultimate  strength  due  to  the  annealing  of  the  finished  eyebar. 

3.  The  steel  is  not  perfectly  homogeneous  and  the  chances  of  a  "soft"  or  weak  place  are  greater  in  a 
large  bar  than  they  are  in  the  small  test  piece. 

As  the  result  of  his  study  of  this  subject,  Mr.  Lewis  recommends  60,000  pounds  per  square  inch  as  the 
minimum  ultimate  strength  of  the  specimen  test  and  56,000  pounds  per  square  inch  as  the  minimum 
ultimate  strength  for  full-sized  steel  eyebars  ;  the  maximum  ultimate  strength  to  be  70,000  pounds  per 
square  inch  in  both  cases. 

2d.  Square  or  Round  Rods  are  used  for  all  members  requiring  sleeve-nuts  or  turn-buckles. 
They  are  the  most  commonly  employed  for  counter-tics,  lateral  and  sway  rods,  and  for  the 
rods  of  Howe  trusses.  They  are  generally  made  of  wrought-iron  when  adjustable,  as  the 
prevailing  belief  is  that  steel  is  weakened  very  much  by  the  sharp  unfilleted  corners  of  the 
screw-threads.  Steel  bars  with  upset  screw  ends  are,  however,  being  made  with  success,  and 
all  rational  objection  to  them  is  disappearing. 

The  various  ways  in  which  rods  are  used  are  shown  in  the  figures  on  pages  248  and  249. 
Fig.  273  shows  the  forged  or  solid  eye,  Fig.  274  shows  the  ordinary  loop  eye,  Fig.  275  shows  the 
rod  with  two  screw  ends  with  either  the  ordinary  nut  or  the  clevis  at  the  ends,  and  Fig.  276 
shows  the  sleeve-nut  or  turn-buckle  detail  which  would  be  used  for  lengthening  or  shortening 
a  rod  with  solid  or  loop  eyes.  Tables  giving  the  standard  dimensions  of  these  various  details 
are  also  given. 

EDGE  MOOR  STANDARD  UPSET  SCREW  ENDS. 


Dimensions  in  Inches. 


SQUARES. 

ROUNDS. 

<  G  ' 

J 

f^-G- 

1 

r 

lillL 

< 

1 

m 

Jiil 

J  

< 

* 

f 

i 

A 

B 

C 

D 

G 

H 

A 

B 

C 

D 

G 

H 

Side  of 

Diameter 

Area  of 

Area  at  Root 

Lengih  of 

Threads 

Diameter 

Diameter 

Area  of  Bar. 

Area  at  Rooi 

I.engih  of 

Threads 

Square. 

of  Screw. 

Bar. 

o(  Tlireail. 

Screw. 

per  Inch, 

of  Bar. 

of  Scicw. 

of  Thread. 

Screw. 

per  Inch. 

1 

.56 

.694 

3f 

7 

f 

7 

•307 

.420 

3i 

9 

7 

li 

.76 

.891 

3l 

7 

* 

I 

•442 

-550 

3i 

8 

li 

1. 00 

1295 

4 

6 

7 

S 

.601 

.694 

3f 

7 

T  1 

If 

1.27 

1.496 

4i 

5i 

.785 

.891 

3f 

7 

li 

1.56 

2  051 

4i 

5 

li 

I« 

■994 

1.057 

4 

6 

If 

li 

2 

l.8g 

2.302 

4J 

4i 

li 

1* 

1.227 

1.295 

4 

6 

2i 

2.25 

3023 

4l 

4i 

If 

If 

1.484 

1-744 

4j- 

5 

If 

2| 

2.64 

3.298 

5 

4 

li 

li 

1.767 

2.051 

4i 

5 

I* 

2i 

3.06 

3-7<9 

5 

4 

If 

2 

2.073 

2.302 

4i 

4i 

I  7 

2f 

3-53 

4.O22 

5i 

4 

If 

2j 

2  405 

2.651 

4f 

4i 

2 

2| 

4.00 

4.924 

5i 

3* 

,7 
Iff 

2i 

2.761 

3.023 

4f 

4i 

3 

452 

5.428 

a 

3i 

2 

2f 

3.141 

3  298 

5 

4 

2i 

3i 

5.06 

6.5 10 

5i 

3i 

2i 

2i 

3.546 

3.719 

5 

4 

2| 

^h^ 

5.64 

7-548 

6 

3i 

2i 

2| 

3.976 

4.159 

5i 

4 

2* 

3i 

6.25 

8.641 

6i 

3 

2f 

2f 

4.430 

4.622 

5i 

4 

2| 

4 

6.89 

9.998 

6i 

3 

2i 

3 

4  908 

5.428 

5* 

3i 

ll 

3i 

5-939 

6.510 

5i 

3i 

3 

3i 

7  068 

7.548 

6 

3i 

3i 

3f 

8.295 

8.641 

6i 

3 

3* 

4 

9.621 

9-993 

6i 

3 

Note.— Area  at  root  of  thread  in  all  cases  greater  than  area  of  bar. 


*  See  Trans.  Am.  Soc.  C.  E.,  1892. 


MODERN  FRAMED  STRUCTURES. 


Squakb  Rods. 

Round  Rods. 

Size  of  Rod. 

Diameter  of  Eye. 

Diameter  of 
Largest  Pin  hole. 

Size  of  Rod. 

Diameter  of 
Largest  Eye. 

Diameter  of 
Largest  Pin  hole 

1  sq. 

1  to  Ij\ 

li  to 

If    to  l/j 
i4    to  If 

Its^  to  I, J 

2  to  2| 
2i      to  2f 

2i     to  2} 

34 
4i 
44 
5 

54 

6 

64 

74 

8 

2i 
24 
2f 
2? 
3J 

3i 
34 
4 
4 

1  diam. 

1  to  I J 
li  to  IS 

'4  lo  '1 
1}  to  1^ 

2  to  2j 
2i  to  2| 
2i  to  2i 

2i 
4i 

5 

54 

6 

64 

74 

8 

li 

24 
2f 

3 

3i 
34 
4 
4 

Side  View  of  Single  toop. 


Bide  VleiY  o£  Eorked.  Xoop 
Fig.  274. 


10 


up) 


il  c 


Upset  Screw-end  Rod  with  Square  or  Hexagonal  Nuts. 


8crew-end  Rod  with  Clevises. 
Fig.  275. 


3(J 


DESIGN  OF  INDIVIDUAL   TRUSS  MEMBERS. 


249 


EDGE  MOOR  STANDARD  CLEVISES. 


 1-  ^x-^ 


o 


I  ! 

!<-T-+  L-  >\ 


B 

E 

L 

P 

T 

W 

X 

Z 

Diameter  of  Screw  Ends. 

Diameter  of 
Eye. 

Length  of 
Fork. 

Diameter  of 
Pin. 

Length  of 
Thread. 

Width  of 
Bar. 

Thickness  of 
Bar. 

Width  in 
Fork. 

Weight. 

1 
li 
I| 
If 
I| 
2i 
2| 
2| 

I 

If 

2 

2i 
2i 
2i 

4 
4 
4 
4 

4j 
5J 
6f 
6| 

5i 

5i 

5i 

5i 

7 

7 

7 

8f 

\\  to  2i 
If  to  2i 
If  to  2^ 
2 

2i 
2i 
2| 
3 

li 
li 
li 
If 
2 

2i 
2i 
2f 

2 
2 
2 
2 
2 

2i 
2i 
3 

i 

h 

f  and  J 
f 

f 

7 

I 

li 
If 
If 
I| 
2i 
2| 
2| 
2j 

5i 
61 
7i 
9 

i3f 

20i 
25i 

The  dimensions  marked  can  be  varied.    Threads  may  be  right  or  left.    Dimensions  in  inches;  weights  in  pounds. 


EDGE  MOOR  STANDARD  SLEEVE-NUTS, 


B 

L 

T 

W 

Diameter  of  Screw. 

Length  of  Nut 
or  Screw  Ends, 

Length  of 
Thread. 

Diameter  of 
Hex. 

Weight 
one  Nut. 

\ 

I 

r  6 

If 

If 

li 

li 

6i 

If 

2 

3 

If 

li 

7 

% 

2f 

4f 

I| 

If 

-5; 
1 

7i 

2tV 

2f 

6f 

1 

2 

8 

2t\ 

3i 

9i 

2i 

2i 

8i 

2i 

3* 

I2i 

2| 
2| 

2i 

9 

2f 

3l 

16} 

2f 

9i 

211 

4i 

214 

2j 

3 

10 

3r\ 
3ff 

4f 

264 

3i 

lOi 

5 

32 

3i 

II 

3f 

5f 

38i 

3f 

Hi 

3fl 

5f 

45 

4 

12 

4,V 

6i 

53i 

li 

12 

2i 

2 

li 

li 

8* 

If 

2 

4 

If 

li 

1  ^ 

9 

I| 
2tV 

2f 

6i 

If  • 

If 

9* 

2f 

8f 

I| 

2 

10 

2lV 

3i 

I2i 

These  nuts  are  forged  with  fibres  in  direction  of  stress  and  are  of  uniform  section.  The  diameter  of  hexagonal 
part  of  any  nut  is  that  of  the  U.  S.  standard  nut  fitting  screw  of  the  larger  diameter,  given  in  the  column  B  of  table. 
Dimensions  in  inches;  weight  in  pounds. 


25° 


MODERN  FRAMED  STRUCTURES. 


For  the  loop  eye,  Fig.  274,  the  diameter  of  the  pin  is  not  linnited.  The  manufacture  of 
the  loop  makes  a  weld  necessary  at  the  point  A.  As  satisfactory  welds  cannot  be  generally 
secured  with  steel,  loop-ended  rods  are  usually  made  of  wrought-iron. 

If  it  were  necessary  to  provide  for  adjustment  in  the  length  of  rods  with  solid  or  loop 
eyes,  as  would  be  the  case  for  counter-ties,  lateral  and  sway  rods,  sleeve-nuts  or  turn-buckles 
would  be  used. 

It  will  be  noticed  that  the  rods  shown  in  Fig.  275  may  be  adjusted  without  using  sleeve- 
nuts. 

The  tables  given  are  the  standards  in  use  by  the  Edge  Moor  Bridge  Works.  Those  used 
by  other  companies  vary  somewhat  in  appearance  and  in  some  of  the  details,  but  those  given 
fairly  represent  the  important  features  of  all.  Some  companies  use  a  larger  diameter  of  upset 
for  a  given  rod  than  that  given  in  the  table.    The  sleeve-nut  or  right  and  left  nut  shown  is 

often  discarded  for  the  open  turn-buckle  shown  in  Fig. 
^   277.    The  advantage  of  the  open  turn-buckle  is  that 
the  ends  of  the  rod  are  visible,  and  it  may  easily  be 
277-  inspected  and  the  positions  of  the  rods  noted.  An 

objection  to  them  is  that  they  may  be  adjusted  by  running  a  bar  through  the  link,  and  are 
thus  liable  to  be  tampered  with  by  incompetent  persons.  An  improvement  which  seems  to 
be  inevitable  in  the  manufacture  of  the  screw  ends  is  the  use  of  smaller  threads. 

3d.  Single  Shapes. — The  difficulty  met  with  in  the  use  of  single  shapes  for  tension 
members  is  to  provide  proper  attachments  or  connections  at  the  ends.  It  is  assumed  that 
connecting  rivets  will  be  so  spaced  as  to  develop  the  required  net  area  of  the  shape  and  at 
the  same  time  produce  no  eccentricity  of  stress  in  the  piece  if  possible.  The  shapes  of 
iron  or  steel  generally  used  for  these  members  are  plates  and  angles.  For  a  plate  the  con- 
nection is  generally  a  single  lap  joint,  which,  while  it  produces  a  small  eccentricity  of  stress, 
usually  developes  the  strength  assumed.  For  maximum  economy  of  metal,  or  maximum 
efficiency  for  the  amount  of  metal  used,  the  net  area  must  be  as  great  as  practicable.  The 
arrangement  of  rivets  shown  in  Fig.  278  {a)  secures  this  result  and  is  preferable  to  that  shown 


LI 


■ — 

0 

000    0  0 

\o  0  o\ 

\ 0  0  o\ 

1 

\o  0  o\ 

{a) 


Fig.  278. 

in  Fig.  278  {b\  as  the  rivets  are  symmetrically  placed  with  respect  cO  the  axis  of  the  piece,  and 
the  cross-section  of  the  plate  is  reduced  by  the  cross-section  of  only  one  rivet-hole. 

For  an  angle  iron,  and  also  for  any  unsymmetrical  shape  iron,  the  great  difficulty  is  to 
9  «  place  the  rivets  so  that  they  will  be  symmetrical  about  the  neutral 

I    o     °     °    O    Q    Q  i  axis  of  the  piece.     Angle  irons  should  always  be  attached  by  rivets 

through  both  legs  to  develop  the  greatest  strength.  Cooper  speci- 
fies that  unless  this  is  done  only  one  leg  of  the  angle  can  be  counted 
as  available  area.    This  detail  is  usually  made  as  shown  in  Fig.  279. 

Bracing  with  riveted  attachments  is  much  stiffer  than  rods 
or  bars  connecting  on  pins.    The  rivets  allow  no  motion  which 
may  occur  with  pin  connections ;  and  besides,  the  elongation  of  the 
piece  itself  is  not  so  much,  as  it  is  proportioned  for  the  net  area, 
Fig.  279.  while  the  stretch  must  result  from  the  elongation  of      gross  area. 


DESIGN  OF  INDIVIDUAL   TRUSS  MEMBERS. 


251 


0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

F 

'I 

0 

0 

0 

E/>-0-0-0- 

-( 

0 

0- 

0 

-e- 

0 

0 

0 

0 

0 

0 

® 

0 

0 

0 

0 

0 

0 

0 

0 

0 

Fig.  280. 


4th.  Compound  Sections,  or  those  composed  of  two  or  more  simple  shapes  riveted 
together,  are  used  for  those  cases  where  a  stiff  ^  ^ 

tension  member  is  wanted.  If  the  end  attach- 
ments are  riveted,  care  must  be  exercised  to  so 
arrange  the  rivets  that  the  stress  is  not  appHed 
eccentrically,  and  also  that  the  largest  net  area 
possible  is  obtained.  If  the  end  attachment  is 
by  means  of  a  pin,  the  ends  of  the  member 
should  be  designed  as  follows.  Fig.  280  shows 
a  compound  section  tension  member.  There 
are  four  conditions  which  must  be  fulfilled,  viz.: 

1st.  The  net  area  of  the  piece,  after  deducting  the  section  cut  out  by  the  rivet-holes 
should  be  the  greatest  possible.    This  would  be  the  net  section  on  the  line  AB  in  Fig.  280. 

2d.  The  net  area  on  the  line  CD  must  exceed  the  net  area  of  the  piece  by  twenty-five 
per  cent.* 

3d.  The  net  area  on  a  line  EF"  between  the  end  of  the  piece  and  the  edge  of  the  pin- 
hole must  be  equal  to  three  fourths  of  the  net  area  of  the  piece. 
4th.  The  bearing  pressure  of  the  pin  must  not  be  excessive. 

In  the  fulfilment  of  any  of  the  above  conditions,  care  must  be  taken  to  so  distribute  the 
rivets  that  the  required  value  of  each  component  piece  is  developed.  Thus,  for  the  net  area 
on  line  CD  the  net  area  of  the  top  angle  iron  is  manifestly  the  gross  area  reduced  by  one 
rivet-hole,  and  the  value  of  the  angle  iron  may  be  determined  either  by  this  net  area  or  by 
the  shearing  value  of  the  number  of  rivets  /ess  one  which  attach  the  angle  iron  to  the  pin 
plates  between  the  line  CD  and  the  end  of  the  piece.  Owing  to  tlie  fact  that  steel  is  not 
fibrous  and  is  approximately  equal  in  strength  in  all  directions,  the  better  practice  is  to  use 
steel  for  the  end  pin  plates  in  compound  tension  members.  Wrought-iron  has  a  strength 
perpendicular  to  its  fibre  of  about  two  thirds  the  strength  parallel  to  the  fibre. 

228.  Compression  Members. — In  Fig.  281  are  given  some  of  the  more  common  forms 
of  compression  members  for  framed  structures.  The  factors  which  usually  determine  the 
form  of  a  compression  member  are  cheapness  of  manufacture  and  the  cost  of  the  com- 
ponent shapes  of  material,  the  efficiency  of  tlic  form  as  a  strut  and  its  suitability  as  regards 
dependent  details.  In  Fig.  281  sections^,  B,  C,  D,  and  E  are  commonly  used  for  top  chords, 
F,  G,  H,  I,  J,  K,  and  L  for  intermediate  posts,  and  AI,  N,  and  O  for  lateral  or  sway  struts.  In 
all  members,  whether  they  are  to  resist  tension  or  compression,  a  symmetrical  section  is  to  be 
preferred,  as  all  question  of  the  eccentric  application  of  the  stress  is  done  away  with.  It  has 
become  the  accepted  practice,  however,  to  use  the  unsymmetrical  sections  above  noted  for  top 
chord  sections.  These  sections  are  stiffer  laterally  than  the  symmetrical  post  sections 
F,  G,  H,  etc.,  and  it  is  due  to  this  fact  that  they  are  preferred  for  chords  and  end  posts,  which 
are  relied  upon  to  transfer  wind  stress  in  bending. 

229.  The  Selection  of  the  Economical  Form  for  a  Compression  Member. — The 
shapes  of  iron  commonly  used  vary  in  price,  so  that  the  designer  should  always  keep  well 
informed  as  to  the  market.  The  engineering  journals  give  very  good  market  reports  from 
which  the  above  information  may  be  obtained.  The  cost  of  the  manufacture  of  the  finished 
piece  is  almost  entirely  dependent  upon  the  quantity  of  rivets  to  be  used  and  the  facility  wit), 
which  they  can  be  driven.  It  will  be  noted  in  this  connection  that  E  (Fig.  281)  requires 
fourteen  lines  of  rivets  ;  A,  C,  D,  F,  and  G  require  eight  lines  ;  B,  H,  I,  J,  and  K  require  four 
lines;  and  that  M  and  iV  require  only  two  lines.  As  it  is  always  preferable  and  cheaper  to 
have  the  rivets  machine-driven,  they  should  be  so  placed  that  they  may  be  driven  with  the 


*  This  is  not  the  universal  practice,  but  is  a  safe  rule.    Some  engineers  malce  the  excess  of  the  area  fifty  per  cent. 


MODERN  FRAMED  STRUCTURES. 


usual  riveting  machine.  Four  lines  of  rivets  in  each  of  the  sections  G  and  /,  or  those  which 
attach  the  lattice  bars,  are  inside  of  the  assembled  piece  and  are  seldom  in  reach  of  a  machine, 
and  are  therefore  usually  hand-driven  and  expensive. 

There  are  no  general  rules  to  guide  the  designer  in  the  selection  of  the  proper  form  other 


J 


L. 


1 
J 


T 
L 


[ 

G 

T 


A 


l^w-iath-*] 

I 

I 

Ju 

M 


Fig.  281 


than  those  gained  by  experience.  The  best  guide,  when  the  necessary  experience  is  lacking, 
is  to  select  that  form  which  will  allow  the  simplest  and  neatest  correct  detail  for  its  con- 
nections. 

230.  The  Economical  Form  of  a  Compression  Member  to  select  to  resist  a  given 
stress  is  dependent  on  the  factors  enumerated  in  the  preceding  article,  particularly  on  the  value 
of  the  form  in  resisting  stress.  The  accepted  formulae  for  determining  the  necessary  area  for 
a  piece  of  given  external  dimensions  introduce  the  least  radius  of  gyration  as  the  only  variable 
when  the  length  of  the  piece  and  its  end  connections,  whether  flat-ended,  one  flat  and  one  pin 
end,  or  two  pin  ends,  are  constant.  The  greater  the  least  radius  of  gyration  the  less  will  be 
the  required  area  to  resist  the  stress.  This  requires  that  such  a  form  be  selected  as  will  give 
the  largest  least  radius  of  gyration.  The  various  forms  shown  in  Fig.  281  differ  in  respect  to 
their  least  radius  of  gyration  even  when  the  width  of  the  section  is  the  same.  Thus,  it  has 
been  found  that  practically  the  least  radius  for  A,  B,  C,  D,  and  E  is  four  tenths  of  the  width, 
for  F,  G,  H,  and  /  it  is  three  eighths  of  the  width,  for  L  it  is  five  sixteenths  of  the  width,  and 
for  M\X.  is  one  quarter  of  the  width.  The  forms  A,  B,  C,  D,  and  E  require  the  least  area  and 
are  further  economical  in  material,  as  they  require  but  one  side  to  be  latticed  and  hence  have 


DESIGN  OF  INDIVIDUAL  TRUSS  MEMBERS. 


253 


less  "  non-effective"  material  than  most  of  the  others.  The  form  L  has  no  latticing,  and  may 
often  be  found  to  require  less  material  on  this  account. 

231.  The  Limiting  Dimensions  of  Compression  Members. — The  natural  result  of 
determining  the  dimensions  of  a  compression  member  by  the  usual  formulae  would  be  to  select 
a  form  giving  the  largest  radius  of  gyration  and  which  would  also  be  one  of  large  width.  If 
this  were  carried  to  extremes,  it  would  result  in  large  members  of  very  thin  metal ;  and  there  is 
manifestly  a  point  at  which  the  iron  would  be  too  thin  to  be  of  any  value  as  a  strut.  The 
practice  is  to  limit  the  thickness  of  material  and  the  dimensions  of  the  piece,  as  shown  in 
Fig.  282. 


Fig.  282. 


The  thickness  of  the  top  plate  should  not  be  less  than  one  fortieth  of  the  distance 
between  the  rivets  connecting  it  to  the  angles  ;  the  thickness  of  the  side  plates  should  not  be 
less  than  one  thirtieth  of  the  distance  between  the  rivets  connecting  it  to  the  angles,  and  the 
angles  should  not  be  thinner  than  three  quarters  of  the  thickness  of  the  thickest  plate  riveted 
to  them.  The  clear  width  between  the  side  plates  should  not  be  less  than  three  fourths  the 
width  of  the  side  plate  for  the  post  section,  and  not  less  than  seven  eighths  of  the  width  for 
the  chord  section  (Fig.  282),  in  order  that  the  radius  of  gyration  about  the  axis  AB  may  be 
equal  to  or  less  than  that  about  an  axis  CD  perpendicular  to  AB. 

The  unsupported  length  of  a  compression  member  should  not  be  greater  than  fifty  times 
the  least  width  of  the  member  or  one  hundred  and  fifty  times  the  least  radius  of  gyration. 
This  is  necessary  in  order  to  secure  stiffness,  which  is  as  desirable  as  strength. 

It  will  be  noticed  that  the  side  plates  in  Fig.  282  must  have  a  thickness  of  at  least  one 
thirtieth  of  the  distance  between  the  rivets  attaching  them  to  the  angles,  while  the  top  plate 
is  limited  to  one  fortieth  of  the  distance  between  rivets.  This  would  only  be  true  of  pieces 
which  are  pin-ended  and  when  the  pins  are  parallel  to  the  axis  AB.  When  the  piece  is 
square-ended  and  receives  its  stress  through  a  butt  joint,  the  thickness  of  all  plates  should 
be  at  least  one  thirtieth  of  the  distance  between  rivets.  The  top  plate  of  pin-ended  members 
is  allowed  to  be  thinner  than  this,  in  order  to  allow  more  metal  to  be  concentrated  in  the  web 
plates,  which  receive  their  stress  directly  from  the  pins,  and  to  secure  a  section  as  symmetrical 
as  practicable.  It  is  conceded  that  the  advantages  thus  obtained  are  more  than  the  disadvan- 
tage due  to  the  use  of  a  thinner  top  plate. 

232.  End  Details  of  Compression  Members. — In  Chapter  XXI  are  given  the  principles 
which  should  govern  the  designer  in  proportioning  the  pin  plates  and  in  deciding  upon  the 
location  and  number  of  rivets  to  use  for  a  pin-ended  piece.  The  student  is  referred  to  that 
chapter,  as  the  principles  can  be  more  readily  understood  when  illustrated  by  a  practicaJ 
example. 

If  the  member  is  square-ended  no  increase  of  the  area  is  necessary,  as  all  that  is  required 
is  that  the  piece  be  held  firmly  in  position,  in  order  to  obtain  an  even  bearing  over  the  entire 
area  of  the  cross-section. 


^54 


MODERN  FRAMED  STRUCTURES. 


In  all  end  connections,  whether  pin  or  square,  arrange  the  rivets  so  that  their  maximum 
efficiency  may  be  realized  and  stagger  them  or  space  them  on  zigzag  lines,  in  order  to  avoid  as 
much  as  possible  the  danger  of  local  failure  at  the  ends  due  to  the  weakening  of  the  plates  by 
the  rivet-holes.  In  some  recent  experiments  on  full-sized  compression  members,  the  pieces 
failed  by  splitting  the  plates  at  the  ends  where  they  were  weakened  by  rivet-holes  before  two 
thirds  of  the  estimated  value  of  the  piece  was  developed. 

Compression  members  theoretically  do  not  require  any  metal  beyond  the  centre  of  the 
pins  at  their  ends,  but  it  is  always  better  to  provide  metal  enough  beyond  the  pin'  to  resist 
displacement  of  the  piece  by  an  external  force.  For  the  posts  of  trusses  and  like  members  it 
is  customary  to  run  the  section  of  the  post  at  least  two  inches  beyond  the  pin. 

233.  Tie  Plates  on  Compression  Members. — There  are  many  methods  of  determin- 
ing the  size  of  tie  or  batten  plates  in  use,  but  none  give  rational  results,  and  the  error  in  all 
cases  is  no  doubt  very  much  on  the  side  of  safety.  As  to  their  thickness  there  is  little  dis- 
agreement, as  all  concede  that  they  should  be  able  to  withstand  either  compression  or  ten- 
sion. In  order  that  they  may  be  able  to  resist  a  compression  stress  the  thickness  must  be  at 
least  one  fiftieth  of  the  distance  between  the  rivet  lines  in  the  segments  which  they  connect, 
or  else  the  plates  may  be  made  of  a  minimum  allowable  thickness  and  stiffened  with  angle 
irons.    These  two  cases  are  illustrated  in  Figs.  283  and  284.    As  to  the  length  of  the  tie- 


\J     ^     \J     u    u  ' 


000000 

Ti£.Plate 

'b""o~cro"'o''a 

0 

0000 

0 

-  "1 

0 

0 

0 

Tie'EIate 

0 

0 

0 

0 

cTo'o'o 

6 

 -iength'  > 

Fig.  283. 


Fig.  284. 


plate  and  the  number  of  rivets  necessary  in  the  connection  with  each  segment,  there  is  a  great 
diversity  of  opinion.  Some  specifications  require  that  their  length  shall  be  one  and  one  half 
times  the  depth  of  the  piece,  while  others  require  the  tie  plates  to  be  square,  and  in  either  of 
these  cases  the  number  of  rivets  used  would  be  determined  by  the  number  which  could  be  put 
in  using  the  minimum  pitch,  usually  three  inches.  Another  specification  which  attempts  to 
be  rational  requires  that  the  tie  plates  shall  be  long  enough  to  take  sufficient  rivets  to  transfer 
one  quarter  of  the  total  stress  on  the  member  from  one  segment  to  the  other.  The  duty 
required  of  the  tie  plate  is  to  hold  the  two  segments  in  line  and  to  take  up  any  bending 
moment  which  an  eccentric  application  of  the  stress  or  an  external  force  may  produce  in  the 
segments.  In  all  compression  members  composed  of  two  channels,  either  solid  rolled  chan- 
nels or  compounded  of  plates  and  angles,  with  the  pin  bearing  on  the  webs  of  the  channels, 
there  is  a  bending  moment  produced  in  each  segment  due  to  the  fact  that  the  neutral  axis  of 
the  channel  is  not  coincident  with  the  centre  of  the  web  of  the  channel.  The  centre  of  the 
application  of  the  stress  is  necessarily  at  the  centre  of  the  web.  This  bending  moment 
should  be  resisted  by  the  tie  plate,  and  the  rivets  connecting  the  tie  plate  to  the  segment 
should  also  be  able  to  transfer  it  from  the  segment  to  the  tie  plate.    The  amount  of  this 


DESIGN  OF  INDIVIDUAL  TRUSS  MEMBERS. 


bending  moment  can  always  be  accurately  determined.  Should  the  member  be  acted  upon 
by  an  external  force  perpendicular  to  the  webs  of  the  channels,  and  if  the  tie  plate  is  placed 
some  distance  from  the  end  of  the  member,  as  is  usually  the  case,  a  bending  moment  is  pro- 
duced in  the  segments,  which  the  tie  plate  and  its  connecting  rivets  should  be  able  to  resist. 
The  closer  the  tie  plate  is  to  the  end  of  the  member  the  less  the  bending  moment  from  the 
latter  cause  becomes,  hence  the  desirability  of  locating  the  tie  plate  as  near  the  end  of  the 
piece  as  practicable.  If  the  tie  plates  were  proportioned  for  the  bending  moment  produced 
by  the  eccentric  application  of  the  stress  at  the  ends  and  in  addition  thereto  for  any  probable 
external  force,  such  as  the  pressure  of  the  wind,  the  result  would  be  more  rational  and  satis- 
factory than  any  of  the  rules  now  in  use.  In  order  to  hold  the  segments  in  line,  all  that  is 
necessary  is  that  the  tie  plates  be  stiffer  against  bending  than  the  segments  which  they 
connect. 

234.  The  Latticing  of  Compression  Members. — There  are  no  rules  other  than  empir- 
ical  ones  in  use  by  which  the  size  and  spacing  of  lattice  bars  for  compression  members  are 
determined.  The  duty  of  the  latticing  is  to  hold  the  segments  composing  the  members 
together  so  that  they  may  act  as  one  piece,  to  resist  any  local  tendency  to  get  out  of  line  or 
to  buckle  and  to  transfer  any  transverse  shearing  force  to  the  ends  of  the  member.  It  is  very 
probable  that  the  first  of  these  conditions  is  fulfilled  when  the  latticing  is  continuous  from 
end  to  end  and  the  individual  bars  have  a  thickness  of  at  least  one  fiftieth  of  their  length,  so 
that  they  may  be  able  to  resist  compression  or  tension.  The  second  condition  is  fulfilled 
when  the  distance  between  the  rivets  attaching  the  lattice  bars  to  one  segment  is  less  than 
sixteen  times  the  width  of  the  segment  in  the  plane  of  the  latticing,  as  experiments  on  pieces 
in  compression  show  that  a  column  whose  unsupported  length  is  less  than  sixteen  times  its 
least  width  will  fail  by  crushing  instead  of  buckling.  The  third  condition  could  easily  be  pro- 
vided for  if  the  shearing  forces  were  known.  Any  probable  external  forces  should  be  pro- 
vided for,  but  in  addition  to  them  there  are  stresses  in  the  bars  due  to  the  bending  of  the 
strut  from  the  direct  stress  which  can  only  be  estimated.  It  has  been  suggested  that,  as  our 
compression  formulae  all  assume  a  certain  extreme  fibre  stress  due  to  the  flexure  of  the  strut, 
from  this  known  extreme  fibre  stress  we  find  an  equivalent  uniform  load  acting  in  the 
plane  of  the  latticing  which  will  produce  this  fibre  stress,  and  from  this  load  find  the  stress  in 
the  lattice  bars. 

As  an  illustration  of  the  above,  assume  the  common  straight-line  formula  for  struts,  9000  —  40  — ,  where 

/=  length  of  strut  in  inches  and  r  =  radius  of  gyration  of  the  strut  about  an  axis  perpendicular  to  the 

plane  of  the  latticing.    The  term  40  —  is  the  extreme  fibre  stress  from  the  bending  moment  induced  in  the 

strut  by  the  direct  stress.  Then  the  uniform  transverse  load,  W,  which  would  produce  this  fibre  stress  in 
the  strut  would  be 


where  A  =  area  of  the  strut  and  b  y.  r  =  distance  of  the  extreme  fibre  from  the  neutral  axis.  If  the  lattice 
bars  were  inclined  at  an  angle  of  45°  with  the  axis  of  the  piece,  the  maximum  stress  on  the  lattice  bars 
would  be 

112^    I  ■X20A 

— 7—  =  -  X  —7—  X  1.4, 

The  width  of  lattice  bars  with  one  rivet  at  each  end  should  be  about  three  times  the 
diameter  of  the  rivet  used. 

When  the  lattice  bars  are  long,  a  saving  in  material  may  be  made  by  using  small  angle 
irons  instead  of  flat  bars,  as  they  are  more  effective  in  resisting  compression  and  hence  require 
less  area  of  cross-section. 

The  common  rules  for  latticing  are  given  in  Art.  271. 


MODERN  FRAMED  STRUCTURES. 


235.  Common  Formulae  for  Compression  Members.— The  formulas  in  common  use 
for  determining  the  proper  area  of  cross-section  for  a  strut  to  resist  a  given  compressive  stress 
may  be  divided  into  two  kinds,  viz.-  ist,  those  that  plot  in  a  curve;  and  2d,  those  that  plot  in 
a  straight  line.  Within  the  limits  of  practical  use  they  will  give  practically  the  same  results, 
and  as  the  straight-line  formulae  are  easier  and  quicker  in  appHcation  they  are  generally 
preferred. 

Rankine's  Formula,  /  =  — -. 

'  ar' 

a  =  i8,cxx)  when  the  strut  has  two  pin  ends ; 

a  =  24,000    "       "     "      "  one  pin  and  one  flat  end  ; 

a  =  36,000    "       "     "      "  two  flat  ends. 

Common  Straight-line  Formula,       p  =  9000  ~ 

d  =  40      when  the  strut  has  two  pin  ends ; 

^  =  35  "     "     "     "    one  pin  and  one  flat  end; 

^  =  30  "     "     "     "    two  flat  ends. 

The  Pennsylvania  Lines  West  of  Pittsburgh  straight-line  formulae : 
Top  Chord  Sections,     p  =  8400  —  84^  • 

Common  Post  Sections,  /  =  8400  —  88^. 

In  all  of  the  foregoing  formulae,  /  =  permissible  stress  per  square  inch,  /  =  length  of  the 
member  in  inches,  r  =  least  radius  of  gyration  of  the  member  in  inches,  and  /i  =  least  width 
of  the  member  in  inches.  The  formulae  used  by  the  Pennsylvania  lines  are  only  to  be  applied 
to  the  chord  and  post  sections  given  in  Fig.  282,  the  general  form  of  their  equation  being 

P  =  8400  ~       whenr  is  dependent  upon  the  form  of  the  member.    These  formulae  also  do 

not  take  into  account  the  differing  end  conditions  of  the  member;  it  being  assumed,  no  doubt, 
by  the  engineers  of  this  road  that  all  members  of  a  truss  really  act  as  pin-ended  members,  and 
that  any  error  in  this  assumption  would  be  on  the  safe  side. 

The  constants  8000,  9000,  and  8400  in  the  foregoing  formulae  are  used  to  determine  the 
permissible  working  stresses  for  railway  bridges  and  are  for  wrought-iron.  It  is  customary 
to  increase  p  25  per  cent  for  highway  bridges.  If  the  material  is  steel,  /  would  ordinarily 
be  increased  20  per  cent  for  railway  bridges  and  50  per  cent  for  highway  bridges. 

The  parabolic  formulae  of  Prof.  Johnson  given  in  (28)  p.  153  are  the  most  rational  yet 
advanced,  and  are  to  be  recommended  for  use  as  they  are  very  easy  of  application  and  are 
the  nearest  approximations  to  the  truth  of  all  the  formulae  we  have. 


DETAILS  OF  CONSTRUCTION, 


257 


CHAPTER  XVIII. 
DETAILS  OF  CONSTRUCTION. 

236.  Riveting. — The  diameters  of  the  rivets  used  in  structural  iron-work  are  \",  f", 

f",  and  \".  Rivets  |"  and  \"  in  diameter  are  most  generally  used,  smaller  sizes  being  used 
only  in  unimportant  details,  or  where  the  clearances  in  the  piece  riveted  prohibit  the  use  of  a 
larger  rivet.  Rivets  larger  than  ^"  diameter  are  only  used  in  pieces  where  the  metal  is  very 
thick  or  where  the  stresses  absolutely  require  larger  rivets.  Field  rivets,  or  those  to  be  driven 
in  the  structure  after  it  is  in  place  by  hand,  should  never  be  larger  than  ^'  diameter,  and 
preferably  f "  diameter,  owing  to  the  difficulty  of  driving  tight  rivets  of  the  larger  size  by 
hand. 

Rivets  are  usually  made  of  the  same  material  as  the  pieces  in  which  they  are  driven.  If 
steel  rivets  are  used,  they  are  made  of  a  very  soft  steel.  Field  rivets  are  generally  of  wrought- 
iron  in  all  cases,  as  the  difficulty  of  driving  good  steel  rivets  prohibits  their  use.  The  range  of 
temperature  at  which  steel  can  be  effectively  worked  is  very  small,  and  as  in  field  riveting 
considerable  time  is  lost  in  passing  the  rivet  from  the  forge  to  the  riveters,  a  rivet  has  time  to 
cool  to  a  point  below  which  it  is  not  advisable  to  do  any  work  upon  it  such  as  would  be 
necessary  in  driving  it. 

237.  Size  of  Rivet-heads. — The  sizes  of  the  heads  of  rivets  vary  slightly.  The  ordinary 
sizes  are  shown  in  Fig.  285. 


-«  d — >- 

 « — a — ►  — 

V 

Fig.  285. 

The  usual  weight  of  a  pair  of  rivet-heads  (information  usually  needed  when  calculating 
the  finished  weight  of  riveted  work)  may  be  obtained  from  the  following  table : 


Size  of  Rivet  in 

Inches. 

Weight  of  One  Pair 
of  Heads  in  Pounds. 

Length  of  Rivet  re- 
quired to  make  one 
Head. 

O.II 

i" 

t 

0.22 

l" 

0.27 

ri" 

1 

0.44 

If 

I 

0.76 

li" 

The  length  of  the  material  required  to  form  one  head  is  also  given  in  the  foregoing  table. 
Rivets  are  furnished  with  one  head  formed,  the  other  head  being  made  when  the  rivet  is 


MODERN  FRAMED  STRUCTURES. 


driven.  In  ordering  rivets  first  find  the  grip  or  distance  between  the  heads  ot  tne  rivet.  The 
grip  of  the  rivet  is  the  thickness  of  the  plates  or  parts  through  which  the  rivet  is  to  be  driven 
plus  of  an  inch  for  each  joint  between  the  plates  to  allow  for  uneven  surfaces  which  prevent 
closer  contact.  This  grip  must  be  increased  in  the  ratio  of  the  area  of  the  hole  to  area  of 
the  rivet  material,  the  hole  usually  being  of  an  inch  larger  in  diameter  than  the  rivet.  To 
this  add  the  length  required  to  form  one  head,  and  the  length  of  the  rivet  under  the  head  is 
obtained.  Thus,  assuming  a  |-inch  rivet  joining  three  half-inch  plates,  the  grip  would  be 
3  X  i  +  tV  —  ^T7-  Increasing  this  in  the  ratio  of  44  to  52  we  obtain  i|,  nearly.  Adding 
inches  for  the  head,  we  get  2\  inches  as  the  length  to  order.  Rivets  are  ordered  in  even 
eighths  of  an  inch. 

For  a  countersunk  head  add  one  half  the  diameter  of  the  rivet  for  the  head. 

Besides  the  full-button  head  and  the  countersunk  head,  in  extreme  cases  rivets  with 
flattened  heads  may  be  used.  These  heads  are  made  by  flattening  the  button  heads,  and 
should  never  be  used  when  the  height  is  less  than  ^  of  an  inch. 

In  calculating  clearances  it  is  always  better  to  provide  for  a  head  one  eighth  of  an  inch 
higher  than  the  standard  head  used  to  allow  for  discrepancies  in  the  length  of  material  used 
and  in  the  upsetting  of  the  rivet.  This  rule  should  apply  to  countersunk  heads  as  well  as  to 
the  other  kinds,  as  the  rivet  often  does  not  upset  sufficiently  to  bring  the  head  flush  with  the 
plates.  Chipping  countersunk  heads  in  order  to  make  them  flush  with  the  plate  is  to  be 
avoided  when  possible  as  it  will  often  loosen  the  rivets,  and  is  expensive  if  done  well. 

Countersunk  rivets  should  never  be  used  in  plates  of  less  thickness  than  one  half  the 
diameter  of  the  rivet.  Also,  as  a  matter  of  economy  in  manufacture,  they  should  not  be  used 
in  long  pieces,  as  the  extra  handling  of  such  pieces  which  would  be  necessary  in  order  to 
countersink  a  few  rivets  would  make  them  expensive. 

238.  Determination  of  the  Size  of  Rivet  to  Use. — The  diameter  of  the  rivet  should  not 
be  less  than  the  thickness  of  the  thickest  plate  through  which  the  rivet  passes,  owing  to  the 
difficulty  of  punching  holes  the  diameters  of  which  are  less  than  the  thickness  of  the  plate. 
While  this  is  not  an  impossibility,  it  is  expensive  owing  to  the  risk  of  tool  breakage.  The  size 
of  the  rivets  is  often  determined  by  the  clearances  or  space  in  which  the  rivet  must  be  driven. 
Shapes  with  small  flanges  will  require  small  rivets,  as  the  practical  limits  of  the  distance  of  the 
centre  of  the  rivet  from  the  edge  of  the  piece,  and  of  the  clearance  necessary  for  the  tool 
in  driving  the  rivet,  then  determine  what  is  the  largest  size  it  is  possible  to  use  (see  Art.  239). 
It  is  customary  and  cheaper  to  use  one  size  of  rivet  throughout  one  entire  piece,  as  a  change 
of  dies  is  avoided  and  the  punching  and  riveting  done  quicker. 

When  none  of  the  foregoing  conditions  fix  the  size  of  the  rivets,  it  is  good  practice  to  use 
that  size  of  rivet  which  will  amply  resist  all  stresses  and  be  most  economical  in  manufacture. 
Smaller  rivets  reduce  slightly  the  cost  of  punching  and  driving  and  weigh  less,  thus  effecting 
a  saving  both  in  the  cost  of  manufacture  and  of  the  material. 

239.  Practical  Rules  governing  the  Spacing  of  Rivets. — The  minimum  distance 
from  centre  to  centre  of  rivets  should  not  be  less  than  three  diameters  of  the  rivet.  Closer 
spacing  than  this  is  liable  to  split  or  otherwise  injure  the  material.  The  maximum  distance 
from  centre  to  centre  of  rivets  in  a  compression  member,  where  it  is  essential  that  the  parts 
riveted  act  as  one  whole,  should  not  be  greater  than  sixteen  times  the  thickness  of  the  thin- 
nest outside  plate  riveted.  In  general,  a  maximum  pitch  of  6  inches  should  not  be  exceeded 
if  it  is  advisable  to  have  the  parts  drawn  sufficiently  close  to  prevent  the  entrance  of  water. 

In  punching  the  rivet-holes  it  has  been  found  that  they  must  be  located  a  certain  distance 
from  the  edge  or  end  of  a  piece,  in  order  to  avoid  all  danger  of  splitting  the  material.  In 
wrought-iron  the  danger  of  splitting  is  less  when  the  weak  section  is  perpendicular  to  the 
fibre  of  the  material  than  when  it  is  parallel  to  the  fibre,  because  of  the  greater  strength  of 
the  material  in  the  direction  of  the  fibre.    Hence  a  hole  may  be  punched  in  wrought-iron 


DETAILS  OF  CONSTRUCTION. 


259 


closer  to  the  edge  than  to  the  end.  Some  empirical  rules  for  these  distances  are  given  in 
Fig.  286. 

B— >j 

Edge 


Fibre 


t= thickness  of  plate  In  inches. 

Minimum  A=K+>^;+HcZ 
Minimum  B=>i+i+Kd 


Fig.  286. 

It  will  be  noted  that  the  minimum  distances  A  and  B  depend  upon  the  thickness  of  the 
plate  and  the  diameter  of  the  rivet,  but  that  the  distance  from  the  edge  of  the  hole  to  the  end 
or  side  of  the  plate  depends  on  the  thickness  of  the  plate  alone.  It  is  well  to  exceed  these 
minimum  distances  whenever  practicable,  a  common  rule  being  to  make  the  end  distance 
twice  the  diameter  of  the  rivets  and  the  side  distance  one  quarter  of  an  inch  less.  It  must  be 
remembered  that  the  diameter  of  the  hole  is  always  one  sixteenth  of  an  inch  larger  than  the 
nominal  diameter  of  the  rivet  to  allow  the  latter  to  be  entered  while  hot.  When  the  rivet  is 
driven  it  is  upset,  and  then  completely  fills  the  hole. 

In  power-riveting,  and  also  in  the  best  hand-riveting,  a  certain  amount  of  room  is  neces- 
sary for  the  tools  holding  or  forming  the  heads.  The  ordinary  requirements  in  riveting  are 
shown  in  Fig.  287. 

By  referring  to  Fig.  287,  it  will  be  seen  that  provision  must  be  made  for  clearing  a  tool 


J— r 


c 


J 


1 


Minimum  A  = 

Fig.  287. 

whose  diameter  is  three  quarters  of  an  inch  greater  than  the  diameter  of  the  rivet-head, 

240.  Length  or  Grip  of  Rivets. — It  has  been  found  that  there  is  a  practical  limit  to  the 
length  or  grip  a  rivet  may  have,  which  if  exceeded  renders  it  almost  iinpossible  to  drive  tiglit 
rivets.  Those  who  have  investigated  this  subject  now  specify  that  the  maximum  grip  of  a 
rivet  shall  never  exceed  four  times  the  diameter  of  the  rivet.  This  is  a  very  essential  factor  in 
good  riveting,  and  should  never  be  overlooked.  Examples  of  bad  designing  in  this  respect 
are  very  common,  particularly  so  in  plate-girder  flanges,  where  a  series  of  thick  flange  plates 
are  riveted  to  the  flange  angles.  Often  the  bearing  plates  on  the  ends  of  a  compression 
member  are  so  thick  as  to  render  the  rivets  which  are  supposed  to  attach  them  firmly  to  the 
main  section  probably  useless.  There  is  always  a  remedy  for  such  cases,  and  it  should  be 
applied. 


26o 


MODERN  FRAMED  STRUCTURES. 


241.  Strength  of  Rivets. — Rivets  may  fail  by  shearing  off,  by  being  crushed,  or  in 
extreme  cases  of  bad  designing  by  flexure.  They  may  also  fail  by  direct  tension,  either  by  the 
head  breaking  off  or,  rarely,  by  failure  of  the  body  of  the  rivet.  The  value  of  a  rivet  in  direct 
tension  is  so  unreliable  that  its  use  to  resist  such  a  stress  is  generally  prohibited. 

The  shearing  stress  per  square  inch  usually  allowed  on  wrought-iron  rivets  is  three 
quarters  of  the  allowed  tensile  stress  per  .square  inch  on  plate  or  shape  iron.  The  following 
table  gives  the  shearing  value  of  the  common  sizes  of  rivets  for  stresses  of  6000,  7500,  and 
9000  lbs.  per  square  inch,  which  are  the  most  common  values  in  use: 

TABLE  OF  SHEARING  VALUE  OF  RIVETS. 


Diameter  of  Rivet  in  Inches. 

Area  of  Rivet  in 

Value  in  Single  Shear  at 

Fraction. 

Decimal. 

Square  Inches. 

6000  Lbs.  per  Square 
Inch. 

7500  Lbs.  per  Square 
Inch. 

9000  Lbs.  per  Square 
Inch. 

\ 

\ 

\ 
7 
F 

I 

0.500 
0.625 
0.750 
0.875 
1 .000 

0.1963 
0 . 3068 
0.4418 
0.6013 
0.7854 

II78 
1841 
2651 
3608 
4712 

1472 
2301 
3313 
4510 
5890 

1766 
2761 

3975 
5412 
7068 

6000  lbs.  per  square  inch  is  usually  allowed  on  railroad  work,  and  7500  lbs.  per  square 
inch  for  roadway  bridges  when  the  material  is  wrought-iron  ;  7500  and  9000  are  the  usual 
corresponding  values  used  for  steel. 

When  the  rivets  are  to  be  driven  by  hand  during  erection  (i.e.,  field  rivets)  it  is  customary 
to  increase  the  number  of  rivets  by  25  to  50  per  cent  to  allow  for  faulty  riveting.  The  diam- 
eter of  the  rivet  is  taken  to  be  that  of  the  rivet  before  it  is  driven.  The  real  diameter  of  the 
rivet  in  place,  if  it  completely  fills  the  hole  and  if  the  holes  in  the  pieces  match  exactly,  is  one 
sixteenth  of  an  inch  larger  than  the  nominal  diameter. 

The  crushing  or  bearing  value  of  rivets  is  usually  estimated  at  so  many  pounds  per  square 
inch  on  the  bearing  area  of  the  rivet  on  the  metal  through  which  it  passes.  This  bearing 
area  is  assumed  to  be  the  diameter  of  the  rivet  by  the  thickness  of  the  piece.  It  is  also 
customary  to  take  the  diameter  of  the  rivet  in  computing  the  bearing  area  as  that  of  the  rivet 
before  it  is  driven. 

The  bearing  or  crushing  values  of  rivets  for  the  various  thicknesses  of  plates  and  the  usual 
allowed  stresses  of  12,000,  15,000,  and  18,000  lbs.  per  square  inch  are  given  in  the  following 
tables : 

TABLES  OF  BEARING  VALUES  OF  RIVETS. 
For  12,000  lbs.  per  square  inch. 


Diameter 
of  Rivet 

in 
Inches. 

Thickness  of  Plate  in  Inches. 

i 

5 

1 

7 

T5 

i 

9 

6 

K 

1 1 

T5 

i 

15 

7 

in. 

i 

i 

i 
7 

I 

1500 

1875 
2250 
2625 
3000 

1875 
2340 
2810 
3280 
3750 

2250 
2810 
3375 
3940 
4500 

2625 
3280 
3940 
4590 
5250 

3000 
3750 
4500 
5250 
6000 

3375 
4220 
5060 
5900 
6750 

3750 
4690 
5625 
6560 
7500 

4125 
5150 
6190 
7220 
8250 

4500 
5625 
6750 
7875 
9000 

4875 
6100 
7310 
8530 
9750 

5250 

6560 

7875 
9190 
10500 

For  15,000  lbs.  per  square  inch. 

in. 

5 

i 

I 

I 

1875 
2340 
2810 
3280 
3750 

2340 
2920 
3510 
4100 
4690 

2810 
3510 
4220 
4920 
5625 

3280 
4100 
4920 
5740 
6560 

3750 
4690 
5625 
6560 
7500 

4220 
5270 
6320 
7370 
8440 

4690 
5860 
7030 
8200 
9380 

5160 
6450 
7740 
9020 
10030 

5620 
7020 
8420 
9840 
II250 

6090 
7610 
9130 
10600 
I2I9O 

6560 
8200 
9840 
I  1490 
I  3120 

DETAILS  OF  CONSTRUCTION. 
For  18,000  lbs.  per  square  inch. 


261 


Thickness  of  Plates  in  Inches. 


i 

s 

Iff 

1 

7 

1 

H 

\ 

1  s 
T5 

7 

2250 

2810 

3375 

3940 

4500 

5060 

5625 

6190 

6750 

7310 

7875 

2810 

3510 

4220 

4920 

5625 

6320 

7030 

7740 

8440 

9140 

9840 

3375 

4220 

5060 

5910 

6750 

7590 

8440 

9290 

10120 

10960 

II8IO 

3940 

4920 

5910 

6990 

7875 

8850 

9840 

10830 

11810 

12800 

13780 

4500 

5625 

6750 

7875 

9000 

10 1 20 

1 1250 

12375 

13500 

14525 

15750 

A  bearing  pressure  of  12,000  lbs.  per  square  inch  is  usually  allowed  on  railroad  work,  and 
15,000  lbs.  for  highway  bridges  where  the  metal  is  wrought-iron.  These  values  would  be 
increased  to  15,000  and  18,000,  respectively,  if  the  material  were  steel. 

As  in  the  case  of  rivets  in  shear,  the  number  of  rivets  required  by  the  above  tables  would 
be  increased  by  from  25  to  50  per  cent  if  the  rivets  were  to  be  "  field"  rivets. 

Rivets  will  fail  by  flexure  only  in  those  cases  of  bad  designing  where  the  rivets  are  long, 
and  it  is  impossible  to  drive  them  tight  enough  to  have  them  upset  and  completely  fill  the 
holes,  or  possibly  in  those  cases  where  an  excessive  thickness  of  pin  plates  is  used  at  the  ends 
of  a  compression  member  and  the  rivets  are  relied  upon  to  transfer  the  bearing  stress  to  the 
main  section  in  a  very  short  distance.    The  latter  case  will  be  understood  from  Fig.  288. 


Fig.  288. 


If  the  number  of  rivets  in  Fig.  288  were  determined  for  single  shear,  and  were  |  in.  in  diameter  and  the 
plates  \  in.  thick,  the  bearing  pressure  which  would  be  equivalent  to  this  number  of  rivets,  allowins  6000  lbs. 
per  square  inch  shearing  stress,  would  be  57,600  lbs.  The  bending  moment  produced  by  this  force  applied 
at  the  centre  of  the  bearing  plates  would  be  57,600  x  ij  =  72,000  in.-lbs.  (assuming  the  web  of  the  channel 
to  be  \  in.  thick).  This  moment  is  either  resisted  by  flexure  of  the  rivets  or  by  direct  tension  on  the  rivets 
near  the  pin.  No  matter  which  way  failure  would  occur,  it  can  readily  be  seen  that  stresses  are  produced 
which  ordinarily  are  not  provided  for,  and  which  may  prove  to  be  important.  Such  details  could  be  im- 
proved by  lengthening  the  pin  plates  and  increasing  the  number  of  rivets,  as  it  would  surely  result  in 
lessening  the  stress  on  the  rivets  from  flexure  and  also  lessen  any  direct  tensile  stress.  Rivets  are  never 
proportioned  for  flexure. 


242.  Kinds  of  Riveted  Joints. — Riveted  joints  may  be  designated  as  of  two  kinds: 
Lap  joints,  as  shown  in  Fig.  289,  and  Butt  joints,  shown  in  Fig.  290. 

Either  of  these  kinds  of  joints  may  be  single-riveted  (i.e.,  one  line  of  rivets),  double- 
riveted  (i.e.,  two  lines  of  rivets),  or  chain-riveted  (i.e.,  more  than  two  lines  of  rivets). 


262 


MODERN  FRAMED  STRUCT  URES. 


The  butt  joint  is  the  one  generally  used,  and  is  the  more  effective  joint,  owing  to  its  sym- 
metry and  the  absence  of  eccentric  stresses.  The  lap  joint  is  only  used  in  unimportant  details 
in  structural  work,  and  should  always  be  discarded  for  the  butt  joint  where  possible. 


I   ^  r\  ^ 


io 

single  Riveted, 


!0 

lo 


o 
o 
o 


Double  Riveted 
LAP  JOINTS, 


Fig.  289. 


0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

Chain  Riveted 


^  ^  


^     ^  ^  


0 

\o 

0 

"0 

1 

0 

— ^  ^3* — ^ 


o 
o 


o  I  o 
o  I  o 
o  lo 


o 
o 


0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

Single  Riveted, 


Double  Riveted 
BUTT  JOINTS, 


Chain  R'ivetea 


Fig.  290. 

243.  Strength  of  Riveted  Joints. — A  riveted  joint  may  fail  either  by  the  rupture  of  the 
rivet  or  of  the  plate  ;  hence  the  maximum  efficiency  is  obtained  where  the  strength  of  the  rivets 
is  just  equal  to  the  strength  of  the  plate.  As  the  shearing  strength  of  wrought-iron  is  about 
three  fourths  of  its  tensile  strength,  it  follows  that  the  shearing  area  of  the  rivets  should  be 
one  third  greater  than  the  net  area  of  the  plate  which  is  in  direct  tension.  Also,  as  the  crush- 
ing strength  of  a  rivet  is  about  one  and  one  half  times  the  tensile  strength  of  wrought  iron,  the 
bearing  area  of  the  rivet  (i.e.,  diameter  of  rivet  by  the  thickness  of  the  plate  on  which  it  bears) 
must  be  two  thirds  of  the  net  area  of  the  plate. 

The  maximum  economy  in  material  is  obtained  where  the  net  area  of  the  plates  or  pieces 
joined  is  the  greatest  possible.    Fig  291  shows  how  this  is  best  accomplished.    It  also  shows 

,  ^  (C:^  1^  <^  X 


— ^7 — ^ 

-3- 


Rivet  holes  1  diameter 


o  o  I  o  d 

o  _     i  _ 


Fig.  291. 

the  best  arrangement  of  rivets  for  a  uniform  distribution  of  the  stress  over  the  entire  area  of 
the  plates  joined.  The  rivets  are  also  placed  symmetrically  about  the  axis  AB  of  the  plate. 
The  plate  X \n  Fig.  291  may  fail  by  tearing  apart  on  the  broken  lines  aic,  dbe,  or  on  the  line 


DETAILS  OF  CONSTRUCTION. 


263 


fbg.  A  failure  on  either  of  the  broken  Hnes  abc  or  dbe  would  be  partly  by  direct  tension  on  a 
section  perpendicular  to  the  fibre  and  partly  by  shearing  parallel  to  the  fibre.  As  the  shear- 
ing strength  of  wrought-iron  parallel  to  the  fibre  is  only  about  one  half  of  its  tensile  strength 
on  a  section  at  right  angles  to  the  fibre,  the  relative  net  area  on  the  broken  lines  abc  and  dbe 
is  less  than  the  length  of  those  lines  diminished  by  the  number  of  rivet-holes  which  are  located 
on  them.  Thus,  for  the  Une  abc  there  would  be  9  —  5  =  4  i riches  of  width  to  be  ruptured  in 
direct  tension,  and  (6  —  2)2  =  8  inches  in  length  to  fail  by  shearing  parallel  to  the  fibre.  As 
the  iron  for  this  kind  of  shear  is  worth  only  one  half  of  what  it  would  be  worth  in  direct 
tension  on  a  section  perpendicular  to  the  fibre,  the  relative  value  of  the  8  inches  along  the 
fibre  is  4  inches  across  the  grain ;  hence  the  relative  net  width  of  the  plate  is  8  inches.  The 
relative  net  width  of  plate  on  the  line  dbe  will  also  be  found  to  be  8  inches,  and  the  net 
width  of  plate  on  the  line  fbg  is  8  inches.  The  net  area  of  the  cover  plates  should  be  equal 
to  the  net  area  of  the  plate  spliced.  This  can  be  accomplished  by  arranging  the  rivets  so  as 
to  get  a  maximum  net  width  of  plate,  and  then  increasing  the  thickness  of  the  cover  plates  if 
necessary  to  obtain  the  required  net  area. 

What  has  been. said  regarding  riveted  joints  refers  particularly  to  joints  in  tension ;  but 
it  applies  fully  as  well  to  compression  joints,  excepting  that  in  the  latter  no  attention  is  paid 
to  the  net  section,  as  it  is  generally  assumed  that  rivets  do  not  weaken  compression  members. 
Close  spacing  of  the  rivets  would  increase  the  danger  of  local  failure  in  compression  members, 
and  should  be  avoided.  In  all  joints  care  should  be  exercised  to  so  arrange  the  rivets  that 
the  stress  may  be  uniformly  distributed  over  the  area  of  the  piece. 


Watertown  Arsenal  Tests  on  Single-Riveted  Double-Butt  Joints. 
I.  open-hearth  steel  plates  i  inch  thick,  {-inch  cover  plates. 

Iron  Rivets,  Machine  Driven.    Holes  Drilled. 

'Strength  lengthwise  =  56,500  lbs.  per  square  inch. 
"       crosswise    =  56,500  "      "      "  " 
Material  of  Plates :     -j  Elastic  limit  =33,000  "      "      "  '* 

Elongation  in  10  in.  —  26  per  cent. 
_  Reduction  of  area    =55  "  " 
(Each  result  the  mean  of  two  tests.    Method  of  failure  indicated  by  bold-faced  type.) 


Size  of  Rivet 
and  of  Hole. 


inch. 
\%  and 


and 


H  and  I 


Maximum  Stress  on  Joint  per  Square  Inch. 


Pitch  of 

Rivets. 

Tension  on  Gross 

Tension  on  Net 

Compression  on 

Shearing  on 

Efficiency  of 

Section  of  Plate. 

Section  of  Plate. 

Bearing  Surface  of 
Rivet. 

Rivet. 

Joint. 

inches. 

If 

39.940 

64,900 

103,800 

38,550 

73.6 

If 

39.420 

61,320 

110,300 

41,060 

72.6 

If 

37,000 

64,800 

86.400 

26,900 

68.2 

I| 

37.740 

62,900 

94,400 

29,500 

69.5 

2 

41.900 

67,050 

111,800 

36,900 

70.1 

2i 

41,000 

63,300 

116, 200 

37.200 

69.7 

I| 

34.600 

64,900 

74,100 

20,000 

63.8 

2 

37,900 

67,400 

86,600 

24,500 

63-4 

38,900 

65,700 

94,400 

25,800 

66.1 

2J- 

38,900 

63,600 

100,000 

27,500 

69.4 

2| 
2* 

39.400 

62,400 

107,000 

29, 300 

71-4 

40,600 

62,500 

116,000 

32,400 

75.6 

2 

34.000 

68,000 

68,000 

16,600 

57.6 

2i 

35.100 

66,400 

74,700 

18,300 

59-6 

2i 

35,600 

64,200 

80, 200 

19,400 

63- 4 

2| 

36,900 

63,700 

87,600 

20,400 

66.8 

2i 

37.400 

62,300 

93.500 

22,400 

69.6 

2| 

39.300 

63,400 

103,300 

24,300 

67.8 

2f 

40,100 

63,100 

110,000 

26,200 

69.4 

2| 

39,800 

61,100 

114,400 

28,000 

74.1 

264 


MODERN  FRAMED  STRUCTURES. 


II.    OPEN-HEARTH  STEEL  PLATES  \  INCH  THICK,  ys-INCH  COVER  PLATES. 

Iron  Rivets,  Machine  Driven.    Holes  Drilled. 

f  Strength  lengthwise  =  58,560  lbs.  per  square  inch. 
I       "       crosswise    =  58,380  "      "      "  " 
Material  of  Plates :    ^  Elastic  limit  =31,600  "      "      "  " 

Elongation  in  10  in.  =  24  per  cent. 
Reduction  of  area    =  50  "  " 
(Each  result  the  mean  of  two  tests.    Method  of  failure  indicated  by  bold-faced  type.) 


Maximum  Stress  on  Joint  per  Square  Inch. 

Size  of  Rivet 
and  of  Hole. 

Pitch  of 
Rivets. 

Tension  on  Gross 
Section  of  Plates. 

Tension  on  Net 
Section  of  Plates. 

Compression  on 
Bearing  Surface  of 
Rivet. 

Shearing  on 
Rivet. 

Efficiency  of 
Joint. 

inch. 
44  and  * 

inches. 
If 

2 

38,400 
39.300 
37,000 

67,100 
65,400 

59-100 

89,500 
98,300 
98,800 

36,600 
40,300 

40,800 

67.1 
68.6 
64.7 

tI  and  I 

I| 
2 

2i 
2i 
2f 

36,900 

37.900 

44,800 
39,800 
40,000 

69,100 
67,400 
76,100 
65,200 

63,400 

79. 100 
86,700 
108,800 
102,500 
108,800 

27.700 
30,600 
37,200 
36,200 
38,500 

64-5 
66.3 
76.0 
66.4 
66.7 

^  and  I 

2 

2i 

2i 
2| 
2* 
2f 
2? 

34.400 
35,000 
36,900 
38,700 
38,700 
39.700 

40,600 

68,800 
66,000 
66,500 
66,800 
64,600 
64,700 
63,900 

68,800 
74,400 
83,200 
92, 100 
96, 800 
104,500 
111,500 

21, 100 
22,400 
25,500 
28,300 
29,800 
31,200 
34,300 

60.2 
59-2 
61.5 
64-5 
66.9 
67.2 
67.6 

ItV  and  i^- 

2i 
2i 
2| 
2* 
2i 
2f 
2i 

3 

3J 

30,400 

33,400 
33,700 

36, 100 
36,400 
38,700 
38,900 
39,900 
38,900 

64,800 
66,800 
63,900 
65,000 
63,700 
65,400 
63,900 
63,900 
60,800 

57.300 
66,800 
71,200 
81,400 
85,000 
94,700 
99,400 
106,300 
108,200 

15,400 
17,600 
19,400 
22,000 
22,700 
25,800 
27,300 
29,100 
29,600 

51-5 
56.5 
58.0 
62.3 
61.6 
64.5 
67.0 
68.8 
67.7 

III.    OPEN-HEARTH  STEEL  PLATES  |  INCH  THICK,  |-INCH  COVER  PLATES. 
Iron  Rivets,  Machine  Driven.    Holes  Drilled. 

C  Strength  lengthwise  =  56,100  lbs.  per  square  inch. 
I       "       crosswise    =  56,700   "     "      "  " 
Material,  of  Plates :     \  Elastic  limit  =28,200   "     "      "  " 

Elongation  in  10  in.  =  27  per  cent. 
[  Reduction  of  area    =  47  "  " 
(Each  result  the  mean  of  two  tests.    Method  of  failure  indicated  by  bold-faced  type.) 


Pitch  of 
Rivet. 


inches. 
2 

2i 


2i 
2| 
2i 

2i 


2^ 
2i 
2f 
2i 
2f 
24 
2| 

3 

3l 


Maximum  Stress  on  Joint  per  Square  Inch. 


Tension  on  Gross 
Section  of  Plate. 


34,100 

35,900 
36,800 


32, 100 
34,100 
36,300 
37,600 
37,100 
36,300 


31, 100 
33,200 
34,500 
34,900 
36,700 
38,100 
38,400 
38,400 
37.900 


Tension  on  Net 
Section  of  Plate. 


64,000 
63,900 
62,600 


64,100 
64,300 
65,200 
65,100 
61,900 
58,800 


66,100 
66,300 
65,600 
63,400 
64,300 
64,600 
63,000 
61,500 
59,200 


Compression  on 
Bearing  Surface  of 
Rivet. 


73,200 

82,  too 

89,400 


64,100 
72,400 
81,600 
89,200 
92,900 
95,300 


58,700 
66,200 
72,800 
77.500 
85,600 
•  93,100 
98,400 
102, 500 
105,000 


Shearing  on 
Rivet. 


33,100 
37,600 

40,600 


25,400 
28,600 
31,800 
36,500 
36,500 
37,600 


20,700 
22,800 
25,200 
27,100 
29,900 
32,100 
33.900 

35.900 
37,100 


Efficiency  of 
Joint. 


61.4 
64.9 
66.9 


58.3 
60.9 
63.2 
65.7 
66.3 
64.7 


56.5 
57-9 
60.2 
62.3 
65-4 
66.5 
67.0 
68.6 
68.9 


DETAILS  OF  CONSTRUCTION. 


265 


IV.    OPEN-HEARTH  STEEL  PLATES  f  INCH  THICK,  ^'s-INCH  COVER  PLATES. 

Iron  Rivets,  Machine  Driven.    Holes  Drilled. 

'  Strength  lengthwise  =  58,300  lbs.  per  square  inch. 
"       crosswise    =  60,200   "     "       "  " 
Material  of  Plates :     \  Elastic  limit  =28,300   "    "  " 

Elongation  in  10  in.  =  27  per  cent. 
Reduction  of  area    =  48    "  " 


Maximum  Stress  on  Joint  per  Square  Inch. 

Size  of  Rivet 
and  of  Hole. 

Pitch  of 
Rivets. 

Tension  on  Gross 
Section  of  Plate. 

Tension  on  Net 
Section  of  Plate. 

Compression  on 
Bearing  Surface  of 
Rivet. 

Shearing  on 
Rivet. 

Efficiency  of 
Joint. 

inch. 
If  and  I 

inches. 
2 

2i 
2i 
2| 

32,000 
34,600 
35.900 
36, 100 

63,900 

66,  TOO 
64.600 
62,200 

64,100 
73.500 
80,800 
85,700 

30,400 
35.800 

38,900 
40,400 

54-2 
58.6 
60.8 
61. 1 

2i 
2j 
2| 
2i 
2f 
2} 
2i 

31, 500 
34,000 

35.300 

38,100 

35,700 
37.700 
39.300 

66,900 
68,000 
67,  too 
69,300 
62,600 
63,800 
64,000 

62,000 
68, 100 
74,700 
84,200 
83,300 
92,300 
93,800 

24,900 
28,700 
32,100 
35,200 
34.700 
38,900 
41,300 

53-4 
57-6 
59.8 
63.0 

59-3 
64.0 
66.5 

and  li 

2i 
2| 
2i 
2| 
2| 
2^ 

3 

3i 
3i 
31 

29,500 
33,600 

34,700 
36,500 
34,500 
38,300 

37,600 

38,300 

36,800 
39,900 

66,400 
71,000 
69,300 
69,400 
63,200 
67,700 
64,600 
63,600 
59.700 
63,300 

53.200 
63.900 
69, 500 
77,200 
75.9"o 
88,200 
90,200 
95,700 
95,400 
107,800 

20,100 
23,800 
26,400 
29,500 
29,200 
35,000 
35.800 
36,500 
36,200 
41,100 

50.0 
55.6 
57-5 
60.4 
58.3 
63-4 
65.0 
64.8 
62.3 
67.5 

U.  S.  Bureau  of  Steam  Engineering  Tests  of  Multiple-riveted  Butt  Joints, 

DOUBLE-RIVETED  BUTT  JOINTS  20  INCHES  WIDE. 
Holes  Drilled.    Rivets  Staggered  and  Machine-driven. 
(Method  of  failure  indicated  by  bold-faced  type.) 


Strength  of 

Plate  in 
Pounds  per 
Square  Inch. 

Maximum  Stress  on  Joint  per  Square  Inch. 

Size  of  Rivet 
and  of 
Hole. 

Pitch  of 
Rivets. 

Distance 
between 
Rows. 

Kind  of 
Rivet. 

Thickness 
of 
Plate. 

Tension  on 

Gross 
Section  of 

Plate. 

Tension  on 
Net  Section 
of  Plate. 

Compression 
on  Bearing 
Surface  of 
Rivet. 

Shearing  on 
Rivets. 

Efficiency 
of  Joints. 

inch, 
f  andU 

inches. 
3 

inches. 

Steel 

inch. 

\ 

53,710 

42,860 
40,960 
42,720 

55.990 
53.530 
55.800 

84,320 
80,690 
84,140 

45.470 

43,000 
42,540 

79.8 
76.2 
79-5 

I  and  i^V 

3f 

2 

Steel 

7 

% 

51,190 

35,180 
36,190 
35.780 

51,150 
52,640 
52,050 

62,560 
64,410 
63,630 

32,340 
33.210 
32.970 

68.7 
70.7 
69  9 

TRIPLE-RIVETED  BUTT  JOINTS  20  INCHES  WIDE. 

f  and  11 

«  9 

If 

Sicel 

1 

53.710 

43,460 
40,390 
44,290 

54.040 
50,200 
55.050 

69,560 
64,630 
70,790 

36,320 
33.410 
36,640 

80.9 
75.2 
82.5 

I  and 

Steel 

7 

51.190 

37,910 
38,400 
37.950 

51,080 
51.720 
51.130 

53,500 
54,190 
53.560 

27,830 
28,420 
27.730 

74.1 
75.0 
74- 1 

1  and  II 

3Tir 

I* 

Iron 

f 

53.710 

43,600 
43,260 
43.000 

54,220 

53.780 

53,460 

69,720 
69,220 
68,820 

35.130 

36,460 
36.390 

81.2 
80.5 
80.1 

266 


MODERN  FRAMED  STRUCTURES. 


244.  Watertown  Arsenal  Tests  of  Riveted  Joints. — In  the  preceding  tables  are  given 
the  results  of  some  tests  on  riveted  joints  at  the  Watertown  Arsenal.  They  agree  practically 
with  others  made  recently  elsewhere  and  are  worthy  of  study,  as  they  show  the  value  of  close 
riveting  in  distributing  the  stress  uniformly  over  the  piece  by  the  higher  values  of  the  net 
sections  where  the  pitch  is  the  least.  They  also  show  the  very  high  crushing  strength  of 
rivets,  their  shearing  strength,  and  seem  to  indicate  that  the  real  strength  of  a  riveted  joint 
is  the  friction  between  the  plates  caused  by  the  tension  on  the  rivets.  Almost  invariably  it 
will  be  found  that  the  tensile  strength  per  square  inch  of  net  area  of  plate  in  the  riveted 
joint  is  greater  than  the  tensile  strength  of  the  specimen  test  of  the  same  material. 

245.  The  Designation  and  Location  of  Rivets  on  Working  Drawings.— Working 
drawings,  or  those  used  in  the  manufacture  of  the  work,  should  always  show  definitely  and 
clearly  the  size,  location,  and  kind  of  the  rivets  used.  As  it  is  customary  to  use  only  one  size 
of  rivet  in  one  complete  piece,  a  single  note  on  each  drawing  giving  the  size  of  the  rivet  and 
of  the  hole  is  generally  all  that  is  necessary.  This  note  should  always  be  prominent  enough 
to  be  easily  found.  The  location  of  the  rivets,  inasmuch  as  this  is  a  question  which  should 
be  determined  by  the  designing  engineer  and  not  by  the  shop  workmen,  can  be  given  by 
dimensions  on  the  drawing.  It  must  be  remembered  that  in  the  manufacture  each  component 
piece  of  a  riveted  member  is  punched  separately,  hence  the  dimensions  must  be  given  so  that 
the  proper  location  of  the  rivets  on  them  will  be  understood.  For  plate  iron  the  rivets  are 
located  from  the  end  and  edge  of  the  piece  as  base  lines.  For  angle  and  Z  iron  the  base  lines 
are  the  ends  and  backs.  For  channel  iron  the  backs  and  end  if  the  rivets  are  in  the  flanges, 
or  from  the  sides  and  end  if  the  rivets  are  in  the  web.  In  general,  for  any  shape  use  as  a  base 
line  an  edge  which  is  definitely  finished  in  the  rolling  of  the  shape  and  never  any  filleted 
corrier  or  rounded  edge.  In  the  following  table  is  given  the  usual  spacing  of  rivets  for  angle 
iron  and  the  distances  to  be  given  in  locating  rivets  for  angles,  channels,  and  I  beams. 


Width  of  Leg 
of  Angle. 

Pitch. 

Double. 

Single. 

W 

y 

2 

X 

1| 

Not 

used 

I 

2 

<< 

If 

(C 

li 

2f 

(( 

(< 

If 

3 

(( 

If 

3i 

(< 

(1 

I| 

3i 

(t 

(( 

2 

4 

<( 

<i 

2i 

4i 

(  < 

o3 
2t 

5 

If 

2 

3 

6 

2i- 

2i 

4 

Besides  giving  the  size  of  the  rivets  and  their  location  it  is  necessary  to  show  on  the 
drawing  whether  they  are  full-button-headed,  flat  headed,  or  countersunk ;  also,  whether  they 
are  to  be  shop-driven  or  are  field  rivets.  This  may  be  done  by  adopting  some  conventiona! 
sign  for  each  of  the  difTerent  kinds  of  rivets.  A  few  years  ago  the  conventional  signs*  shown 
on  page  267  were  generally  adopted  by  the  bridge  companies  and  consulting  bridge  engineers, 
and  it  would  save  a  great  amount  of  annoyance  if  they  were  universally  adopted.  They  have 
stood  the  test  of  continual  use  for  two  years,  and  are  to  be  recommended  for  their  simplicity 
and  clearness. 


*  Originally  devised  by  Mr.  Frank  C.  Osborne,  C.E. 


DETAILS  OF  CONSTRUCTION. 


267 


The  heads  marked  a  are  countersunk,  but  are  not  chipped  so  as  to  bring  them  flush  with 
the  surface  of  the  plate.  It  is  impossible  to  determine  the  exact  length  of  material  required 
for  countersunk  rivets  which  will  result  in  tight  rivets  and  also  drive  flush  with  the  plate.  The 
heads  marked  b  and  c  are  button-heads  flattened. 


o 


D 


0 


0 


( 


) 


) 


(ID 
0 


0 


(::: 


6- 


6- 


h 
) 

if 


) 


# 


m 


o 


C::) 


Maximum  height  of  heads  marked  a  =  %  Inch. 
"  ■'      "    ••  '>      c  =  %  " 

Conventional  Rivet  Signs, 

246.  Bolts  are  manufactured  either  "  rough "  or  "finished."  The  finished  bolt  is  the 
rough  bolt  finished  to  exact  dimensions.  Rough  bolts  are  used  in  the  temporary  fitting  up 
of  work  in  the  shops  and  during  erection,  and  generally  for  all  woodwork.  Finished  bolts  are 
very  expensive,  and  are  only  used  in  those  cases  where  a  close  fit  is  absolutely  essential.  The 
latter  are  often  used  as  a  substitute  for  rivets,  in  which  case  they  are  proportioned  for  the 
same  allowed  stresses  as  rivets.  In  cases  where  rivets  would  be  subjected  to  direct  tension 
tending  to  pull  off  the  rivet-heads,  finished  bolts  are  now  generally  used,  as  they  are  more 
reliable  than  rivets  to  resist  such  stresses.  '  Where  finished  bolts  are  to  be  used  it  must  also  be 
borne  in  mind  that  the  holes  for  them  must  be  drilled  to  an  exact  fit  with  the  bolts,  and  that 
this  adds  to  the  expense.  An  advantage  of  the  use  of  bolts  in  place  of  field  rivets  is  that  the 
work  can  be  done  much  quicker,  and  when  it  is  desirable  to  erect  a  span  as  quickly  as  possible 
bolts  may  be  used  to  advantage. 

When  ordering  bolts  give  the  diameter,  length  under  the  head,  and  length  of  thread  neces- 
sary.   Tables  giving  the  standard  dimensions  of  bolts  may  be  found  in  various  handbooks. 

247.  Anchor  Bolts. — The  ordinary  anchor  bolts  which  are  used  to  hold  the  bed  plates 
of  a  span  in  position  are  rough  bolts  and  are  made  in  various  styles.  The  three  styles  shown 
in  Fig.  292  are  the  ones  most  commonly  used. 

The  wedge  bolt,  {a)  Fig.  292,  is  split  at  the  bottom  and  a  small  wedge  inserted,  which 
when  the  bolt  is  driven  in  expands  the  lower  end  of  the  bolt  and  prevents  its  being  pulled 
out.  The  "  rag  "  or  "  swedged  "  bolt  {b)  has  indentations  on  its  surface.  The  bolt  {c)  has  a 
screw  thread  cut  on  its  lower  end.  When  the  bolts  are  put  in,  the  hole  is  filled  with  sul- 
phur, lead,  or  cement,  which  sets  and  holds  the  bolt. 


268 


MODERN  FRAMED  STRUCTURES. 


Where  the  anchor  bolts  are  not  relied  on  to  resist  any  direct  tension,  they  are  put  from  six 
to  nine  inches  in  the  masonry. 

If  the  bolts  are  relied  on  to  take  a  direct  tensile  stress — as,  for  instance,  at  the  foot  of  a 
high  trestle  post  or  for  elevated-railroad  posts, — they  are  put  in  the  masonry  as  far  as  it  is 
necessary  to  insure  a  weight  of  masonry  resting  on  them  sufficient  to  resist  the  tension  on 


Fig.  292. 


the  bolt.  These  bolts  are  usually  provided  with  a  head  on  their  lower  end,  which  bears  on  an 
iron  plate.  If  made  in  this  way  they  are  built  into  the  masonry  when  the  latter  is  laid.  Bolts 
with  screw-ends,  as  shown  in  (^r),  Fig.  292,  have  been  set  in  masonry  with  cement,  and  when 
tested  have  developed  the  value  of  the  bolt  in  tension. 

248.  Pins. — The  following  table  gives  the  standard  sizes  of  pins  made  by  the  Edge  Moor 
Bridge  Works.  The  diameter  of  the  finished  pin  is  given  in  sixteenths  of  an  inch,  as  the 
material  from  which  the  pin  is  made  is  furnished  in  sizes  measured  in  even  eighths  of  an  inch, 
and  one  sixteenth  of  an  inch  is  taken  off  in  turning  or  finishing  the  pin.    The  table  also  gives 

STANDARD  TRUSS  PINS  AND  NUTS. 


SOLID  NUT 

LOMAS  NUT 

PILOT 

Diameter  of  Pin  =  A. 

Thread. 

Solid 

Nut. 

Lomas  Nut. 

Nominal. 

Finished. 

Diameter  =  B. 

Length  =  C. 

Short  Diameter 
=  E. 

Long  Diameter 
=  F. 

Depth  of 
Recess  =  G. 

Short  Diametei 
=  H. 

Long  Diameter 

in. 
2\ 
2f 

3 

3i 
3i 
3i 
4 

4i 
4i 
4* 
5 

5i 

5i 

5f 

6 

6i 

6i 

6f 

7 

in. 

3t\ 
3A 
3ii 

ol  6 

4Tff 

4A 

4Ti 

4tI 
6H 

in. 
2 
2 
2 

2i 
2i 
2f 

3 

3i 

3i 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

in. 
If 

t  4 
tt 
€t 

tt 
€« 
tt 
1 1 

«e 
tt 
tt 
tt 
1 1 
tt 
tt 
tt 

in. 

3 

3i 

3i 

4 

4 

4i 

4i 

5 

5 

Si 

Si 

6 

6 

61 

61 

7 

7 

71 
71 

in. 

3i 

4tV 

4tV 

4\l 

4x5 
c  3 

c  3 

5if 
511 
6| 
6f 

Al  5 

"ts 

6if 

71 

71 

8i 

8i 

1 

in. 
1 

tt 

te 

4  c 

8 

it 
tt 

t< 

<4 
4  ( 
tt 

t  t  ■ 
(t 

tt 

in. 

3i 

31 

3f 

4 

4i 

4i 

4f 

5 

5i 

5f 

5f 

6 

ti 

6f 

6f 

7 

7i 

7f 
71 

in. 
3l 
4tV 
4* 
4U 
4H 
Si 
Si 
5il 
6tV 
6H 
6H 

611  ■ 

7i 

7ll 

711 

H 

8f 

8H 

811 

Note. — The  length  of  thread  for  pins  with  Lomas  nuts  is      inches  for  all  pins  less  than  3^^  diameter. 


DETAILS  OF  CONSTRUCTION. 


269 


the  standard  sizes  for  the  solid  pin  nut  and  the  Lomas  pin  nut.  The  Lomas  nut  is  now  the 
one  most  commonly  used  ;  the  recess  in  the  nut  allowing  the  nut  to  be  drawn  up  tight  against 
the  bars  packed  on  the  pin,  as  there  are  usually  small  inaccuracies  in  the  thickness  of  the  bars 
which  prevent  an  exact  calculation  of  the  grip  of  the  pin.  Before  the  Lomas  nut  came  into 
use  a  wrought-iron  washer  bored  to  fit  the  pin  was  used  with  the  solid  pin  nut  to  accomplish 
the  same  purpose.  The  pilot  nuts  shown  are  put  on  the  pin  to  protect  the  thread  and  assist 
in  guiding  the  pin  while  it  is  being  driven. 

In  calculating  the  grip  of  pins  it  is  customary  to  increase  the  net  grip  ^  of  an  inch  for 
each  eyebar,  and  one  fourth  of  an  inch  for  each 
riveted  compound  member.  Thus  in  Fig.  293 
the  net  grip  would  be  27  inches  but  to  allow  for 
inaccuracies  in  the  thickness  of  the  members,  the 
grip  of  the  pin  would  be  made  2/^  iches.  For  the 
same  reason,  when  there  are  several  members 
packed  inside  of  a  built  section,  as  for  instance  in 
a  top  chord,  the  clear  width  inside  of  the  section 
must  be  enough  to  allow  the  eyebars  to  be  ^-^  of 
an  inch  thicker  than  their  nominal  thickness  and 
the  posts  or  built  sections  \  of  an  inch  wider  than 
their  net  width,  and  besides  this  the  net  clear 
width  of  the  chord  must  be  taken  as  \  of  an  inch 
less  than  the  nominal  net  width. 

The  members  on  a  pin  should  be  packed  as 
closely  as  possible.    When  it  is  necessary,  in 
order  to  reduce  the  bending  moment  on  the  pin, 
to  pack  eyebars  in  pairs,  they  should  be  separated  *93- 
at  least  one  half  of  an  inch,  to  prevent  the  collection  of  dirt  and  water  and  to  allow  the  bars 
to  be  painted. 

249.  Lateral  Pins. — The  pins  used  in  the  lateral  system,  if  large,  are  the  same  as  those 
 1^    used  for  truss  pins.    If  they  are  small  or  under  2\  inches  in  diameter,  the 


Fig.  294. 


cotter  pin  shown  in  Fig.  294  is  used.  Sometimes  both  a  nut  and  cotter  are 
used  on  this  style  of  pin.  Lateral  pins  should  always  be  packed  so  that 
they  will  be  in  double  shear.  This  point  is  mentioned  because  it  was 
formerly,  and  to  a  certain  extent  is  now,  the  practice  to  use  them  in  single 
shear. 

250.  The  Calculation  of  the  Stresses  on  Pins. — A  pin  must  be  analyzed  as  a  short 
beam  which  is  subjected  to  excessive  shears  and  bending  moments.  It  must  be  dimensioned 
for  three  kinds  of  failure,  viz. : 

1.  Shearing. 

2.  Cross-bending. 

3.  Crushing. 

Dimensioning  for  Shear. — It  is  customary  to  regard  the  shearing  stress  as  uniformly  dis- 
tributed over  the  cross-section,  in  which  case  the  intensity  of  the  shearing  stress  is 


s  = 


7tr 


(0 


But  from  eq.  (10),  Chapter  VIII,  we  obtain  for  the  intensity  of  the  shearing  stress  on  the 
neutral  plane, 

4       4  ^ 


3  ^r' 


(2) 


MODERN  FRAMED  STRUCTURES. 


in  other  words,  for  a  solid  circular  section  the  maximum  shearing  stress  is  four  thirds  the 
mean  stress.  A  sufficiently  low  working  intensity  of  the  shearing  stress  is  always  taken  to 
allow  equation  (i)  to  be  used  with  safety,  or  to  assume  that  the  shearing  stress  is  uniformly 
distributed. 

For  Wrought-iron  Pins  use  for  mean  intensity  of  shearing  stress  in  pins  8000  lbs.  per 
square  inch. 

For  Steel  Pins  use  10,000  lbs.  per  square  inch  for  shear. 

The  maximum  shearing  stress  at  any  section  is  found  by  making  a  continued  sum  of  the 
horizontal  and  of  the  vertical  components  of  the  forces  coming  upon  it.  The  maximum 
shear  at  any  section  is  the  square  root  of  the  sum  of  the  squares  of  the  horizontal  and  vertical 
shears  at  that  section. 

251.  Bending  Moment  on  Pins. — If  a  pin  be  regarded  as  a  beam,  the  bending  moment 
at  any  section  may  be  found  by  assuming  the  loads  to  be  concentrated  at  the  centres  of  the 
bearings  of  the  members  meeting  at  that  joint.  So  long  as  the  pin  remains  unbent  this  is 
correct.  When  the  pin  bends  appreciably  the  members  no  longer  bear  evenly  upon  it,  and 
the  above  assumption  may  become  very  erroneous,  since  both  the  forces  and  the  lever  arms 
change  as  a  result  of  the  bending,  and  the  bending  moment  becomes  very  much  less  than 
would  appear  from  the  computation. 

The  members  should  be  packed  on  the  pin  in  such  a  way  as  to  produce  the  least  moment. 


s 

3  I  4  2  o  3  4  I  3 


o 


Fig.  295.  Fig.  296. 

An  apparently  slight  change  in  the  arrangement  produces  astonishing  changes  in  the  maximum 
moment.  Thus  in  Fig.  295,  which  is  an  actual  case,*  we  have  the  following  computation  of 
the  shears  and  bending  moments : 


•Reported  by  Mr.  Frank  C.  Osborn,  C.E,,  in  Engineering  Ne^vs,  Feb.  18,  1888. 


DETAILS  OF  CONSTRUCTION. 
HORIZONTAL  MOMENTS  AND  SHEARS,  FIG.  295. 


271 


Bend 

iiig  Moment. 

Member 

Stress. 

Shear. 

Lever-arm. 

Increment. 

Total  Moments  in 
Inch-pounds. 

^1 

-j-  45,600 

—  30,200 
+  45)6oo 

—  30,200 

—  30,800 

-f-  45,600 
+  15,400 
-j-  61,000 
+  30,800 
0 

1  inch 
I  *' 
1  ** 

jf " 

45,600 
I5i400 
61,000 
42,300 

45,600 
61,000 
122,000 
164,300 

HORIZONTAL  MOMENTS  AND 

SHEARS,  FIG.  296. 

Bending  Moment. 

Member. 

Stress. 

Shear. 

Lever-arm. 

Increment. 

Total  Moments  in 
Inch-pounds. 

L, 
Ri 

R* 

—  30,200 
+  45i6oo 

—  30,200 
-j-  45,600 

—  30,800 

—  30,200 
-f  15,400 

—  14,800 
+  30,800 

0 

I  inch 
1  " 
I  " 
>A  " 

—  30.200 
+  i5>400 

—  14,800 

4-  48,100 

—  30,200 

—  14,800 

—  29,600 
+  18,500 

In  this  case  the  vertical  components  of  the  shears  and  moments  would  not  materially  add 
to  those  from  the  horizontal  components  and  are  not  computed.  It  is  evident  at  once  that 
the  second  arrangement,  shown  in  Fig.  296,  gives  less  than  one  fifth  as  great  a  bending 
moment  as  the  actual  arrangement  shown  in  Fig.  295. 

The  bending  moment  may  also  be  found  graphically,  as  shown  in  Fig.  295,  by  means  of 
bending  moment  diagrams.* 

//  My, 

2'>2.  Fibre  Stress  in  Pins. — From  the  formula  M  —  —  or  f=  — ,  we  have  for  a  solid 
cylindrical  beam 

^     Mr           M  ^  ^ 

=^°-^^  <3) 

-r 
4 

In  the  above  example  the  pin  was  2\  inches  in  'iameter;  hence  we  have,  as  the  stress  on  the 
extreme  fibre  with  the  first  arrangement, 


10.2  X  164,300 


=  108,000  lbs.  per  square  inch. 


Evidently  there  was  no  such  fibre  stress  in  the  pin  or  it  would  have  broken.  It  doubtless 
was  considerably  bent  under  this  load,  and  this  bending  caused  a  redistribution  of  the  eye-bar 
stresses  in  such  a  way  as  to  greatly  diminish  the  bending  moment  on  the  pin,  but  with  the 
effect  of  overstraining  some  of  fhe  bars.    The  pin  can  only  perform  its  function  of  trans- 


*  By  obtaining  the  vertical  and  horizontal  forces  acting  on  the  pin,  and  constructing  two  moment  diagrams  on  a 
line  at  45°  with  these  directions,  the  vertical  forces  forming  a  diagram  with  vertical  ordinates,  and  the  horizontal  forces 
one  with  horizontal  ordinates,  the  actual  maximum  moment  may  be  taken  off  with  a  pair  of  dividers  by  setting  on 
corresponding  portions  of  these  two  diagrams.    The  authors  prefer  the  tabular  computations,  however,  as  given  above. 


372  MODERN  FRAMED  STRUCTURES. 

mitting  the  stresses  and  distributing  them  equitably  amongst  the  members  by  remaining 
sensibly  straight.  In  Chapter  VIII  it  was  shown  that  the  computed  stress  on  the  extreme 
fibre  of  a  bent  beam  always  exceeds  the  actual  stress  on  these  fibres  when  bent  beyond  the 
elastic  limit.  In  this  case,  although  the  pin  did  not  break  it  was  evidently  greatly  over- 
strained. 

TABLE  GIVING  FACTORS  WHICH  IF  MULTIPLIED  BY  THE  BENDING  MOMENT  IN  INCH-POUNDS 
ON  A  PIN  WILL  GIVE  THE  OUTER  FIBRE  STRESS  ON  THE  PIN  IN 
POUNDS  PER  SQUARE  INCH. 

I0.2 

Formula  :    /  =   M. 


Diameter  of  Pin  in  Inches. 


2 

2i 

2i 

2? 

3 

3i 

3i 

3i 

4 

4i 

4i 

4f 

5 

5i 

6 

I0.2 

Factor 

1.275 

0.900 

0.654 

0. 490 

0.378 

0.297 

0.238 

0.193 

0.159 

0.133 

0.112 

0.004 

0.082 

0.061 

0.047 

Maximum  Permissible  Bending  Moments  on  Pins. 


Extreme  Fibre  Stress  per  Square  Inch.  • 

Diameter  of 

Pin. 

15000 

18000 

20000 

21000 

22500 

It\ 

2470 

2960 

3290 

3450 

3700 

4380 

5250 

5830 

6130 

6560 

IH 

7080 

8500 

9430 

9910 

10620 

lit 

10710 

12860 

14280 

15000 

16070 

2t5 

15420 

18500 

20550 

21580 

23130 

2A 

21330 

25600 

28430 

29850 

32000 

2H 

28590 

34310 

38  roo 

40050 

42890 

2t5 

37J40 

44800 

49760 

52250 

56000 

3Tff 

47700 

57240 

63570 

66750 

71560 

3A 

3li 

3lf 

59830 

718CC 

79740 

83750 

89750 

73860 

88630 

98430 

103400 

I 10800 

89920 

107900 

II 9800 

125900 

134900 

4Tff 

108200 

129800 

144100 

15 1400 

162200 

4tV 

128700 

154500 

171500 

180200 

I 93 100 

4H 

151700 

182100 

202200 

212400 

227600 

4t5 

177300 

212800 

236300 

248200 

266000 

SA 

205600 

246S00 

274000 

287800 

308400 

236800 

284200 

315600 

331500 

355200 

Sit 

271000 

325200 

361100 

379300 

406500 

r  1 5 

308300 

370000 

410900 

431600 

462500 

348900 

418700 

465000 

488400 

523400 

393000 

47x600 

523700 

550000 

589400 

Oyr 

440500 

528700 

587100 

616600 

660800 

611 

491800 

590200 

655400 

688500 

737700 

7^ 

545830 

655000 

727800 

764200 

818700 

7^ 

605900 

727000 

807800 

848200 

908800 

In  the  case  of  suspension-bridge  pins  where  nearly  the  same  total  stress  is  transmitted 
continuously  in  one  plane,  the  transverse  load  being  relatively  very  small,  the  stresses  in  these 
may  be  omitted  in  finding  the  stresses  on  the  pins.  There  are  then  three  methods  of  arrang- 
ing a  series  of  eyebars  on  a  pin,  as  shown  in  Figs.  297,  298,  and  299.  In  the  first,  Fig.  297, 
the  bars  are  all  the  same  size,  and  alternate  as  shown.  In  the  second.  Fig.  298,  they  are  of 
the  same  size  and  are  arranged  in  pairs.  In  the  third,  Fig.  299,  the  two  outer  bars  are  one 
half  the  width  of  the  others,  and  are  arranged  alternately. 

In  the  first  case  the  bending  moment  increases  regularly  from  the  end  to  the  centre,  each 
pair  of  bars  forming  a  couple  the  moments  of  which,  on  either  half  of  the  pin,  are  all  of  the 
same  sign.  The  maximum  bending  moment  on  the  pin  is  here  nPt,  where  n  is  one  half  the 
number  of  bars  coming  to  the  pin  from  one  direction,  or  n  =  number  of  couples  on  one  half  of 
the  pin;  P—  stress  in  one  bar;  and  t  =  thickness  of  one  bar  plus      in.  for  probable  spacing. 


DETAILS  OF  CONSTRUCTION. 


«73 


In  the  second  case,  the  bars  being  in  pairs,  the  greatest  moment  is  that  due  to  but  one 
couple,  since  this  moment  is  at  once  reduced  to  zero  by  another  couple  of  the  opposite  sign. 
Or  the  maximum  moment  here  is  Pt. 

In  the  third  case,  the  outer  bars  being  half  the  thickness  of  the  others,  the  moment  is 
still  less,  as  shown  by  the  moment  diagrams  accompanying  the  several  arrangements. 

The  arrangement  in  pairs.  Fig.  298,  can  often  be  used  in  packing  lower  chord  joints  in 
truss  bridges.  A  Httle  intelligence  and  care  in  fixing  the  arrangement  of  these  members  on 
the  pin  may  prevent  very  serious  blunders  when  it  comes  to  erection.  The  exact  arrange- 
ment should  always  be  clearly  indicated. 


Fig.  297.  Fig.  298.  Fig.  299. 


253.  The  Bearing  or  Crushing  Value  of  Pins  is  taken  as  so  many  pounds  per  square 
inch  on  the  bearing  area  of  the  pin  on  the  bearing  plates.  This  area  is  found  by  multiplying 
the  diameter  of  the  pin  by  the  thickness  of  the  bearing.  The  usual  pressure  allowed  is  12,000 
pounds  per  square  inch  for  railway  and  1 5,000  pounds  per  square  inch  for  highway  construction 
if  either  the  plates  or  the  pin  or  both  are  of  iron.  If  both  the  bearing  and  the  pin  are  of  steel, 
15,000  and  18,000  pounds  per  square  inch  are  the  corresponding  values  which  are  commonly 
used. 

The  pressure  between  the  bearings  of  eyebars  and  pins  is  usually  not  limited  except  by  a  clause  in  the 
specifications  which  limits  the  niinimmn  diameter  of  the  pin  to  three  fourths  the  width  of  the  bar.  When 
this  minimum  pin  is  used  the  bearing  pressure  per  square  inch  is  33^  per  cent  greater  than  the  tensile  stress 
per  square  inch  on  the  bar,  which  is  not  an  excessive  ratio. 

The  diagram  given  on  page  274  gives  the  bearing  value  of  pins  on  plates  of  different  thick- 
nesses allowing  a  pressure  of  12,000  pounds  per  square  inch.  Similar  diagrams  allowing  15,000 
and  18,000  pounds  per  square  inch  should  be  made  for  practical  use. 

254.  The  Bearing  Strength  of  Rollers.* — When  a  cylindrical  roller  bears  on  a  flat 
plate,  or  transmits  a  load  when  resting  between  two  flat  beds,  the  linear  element  in  contact 
for  a  zero  pressure  becomes  a  surface  of  considerable  width  as  the  pressure  increases,  since 
the  materials  of  both  bed  and  roller  are  elastic.  The  law  of  the  distribution  of  the  stress 
from  this  small  contact  area  over  the  cross-section  of  roller  and  plates  is  unknown.  The 


*  The  discussion  contained  in  this  and  the  following  article  is  taken  from  a  paper  contributed  by  Prof.  Johnson  to 
the  Engineers'  Club  of  St.  Louis,  December,  1S92. 


MODERN  FRAMED  STRUCTURES. 


DETAILS  OF  CONSTRUCTION.  275 

intensity  of  this  stress,  too,  is  always  far  beyond  what  is  ordinarily  known  as  the  elastic 
limit.  This  limiting  stress  corresponds  to  the  maximum  distortion  which  will  entirely  dis- 
appear when  the  load  is  removed,  or  to  the  initial  permanent  set.  This  permanent  set  or 
fixed  distortion  is  a  kind  of  cold  flowing  of  the  material.  When  a  plain  cylindrical  column 
is  subjected  to  a  uniform  compressive  stress  over  its  entire  cross-section,  as  in  Fig.  300,  it 
•nay  be  said  to  be  in  a  condition  of  free  fiow,  since  it  is  free  to  spread  in  all  directions 


Fig.  300. 


Fig.  301. 


Fig.  308. 


throughout  the  length  of  the  column.  In  Fig.  301  the  material  is  compressed  uniformly  over 
a  small  area,  as  with  a  die.  Here  there  is  a  flowing  of  the  metal  laterally,  and  then  vertically, 
finding  escape  around  the  edges  of  the  die.  This  is  a  condition  of  confined  or  restricted  flow, 
and  evidently  the  elastic  limit  here  will  be  much  higher  than  with  the  simple  column.  In  Fig. 
302  the  surface  is  compressed  by  a  cylinder,  or  sphere,  the  greatest  distortion  being  at  the 
middle  of  the  area  of  contact.  When  this  metal  is  forced  to  move,  or  flow,  it  can  find  escape 
only  out  around  the  limits  of  the  compressed  area.  But  at  these  limits  the  metal  is  very 
little  compressed,  and  hence  must  be  moved  from  the  centre.  The  confining  ring  of  metal 
inside  the  limits  of  external  flow  is  now  much  wider  and  hence  the  resistance  to  flow  much 
greater,  so  that  this  condition  will  be  found  to  have  a  higher  elastic  limit  stress  than  that 
shown  in  Fig.  301,  and  very  much  above  the  ordinary  "  elastic  limit  in  compression,"  which  is 
found  for  the  free-flow  condition  of  Fig.  300. 

What  the  relation  between  stress  and  strain  is  in  the  case  of  a  roller  on  a  plane  surface  is 
also  unknown.  It  is  evidently  not  that  given  by  the  ordinary  modulus  of  elasticity,  called 
Young's  modulus,  which  gives  the  relation  for  uniform  direct  stress,  as  in  Fig.  300. 

Therefore,  since  we  do  not  know  the  elastic  limits  for  rollers  on  planes,  and  have  no 
knowledge  of  the  particular  distribution  of  the  stress,  or  of  the  modulus  of  elasticity,  under 
such  conditions,  it  is  evident  that  we  are  not  in  a  condition  to  make  a  theoretical  analysis  of 
the  problem.  All  such  analyses  as  have  been  made  of  this  class  of  problems,*  as  of  bearing 
rollers,  of  wheels  on  rails,  which  is  the  case  of  two  cylinders  crossing  at  right  angles,  afid  of 
spheres  on  planes,  have  been  made  on  very  violent  assumptions,  and  the  conclusions  are  far 
from  agreeing  with  the  facts  as  shown  by  experiment. 

255.  The  Problem  Solved  Experimentally. — In  order  to  obtain  data  for  solving  the 
problem  of  the  bearing  strenjrth  of  rollers  experimentally.  Profs.  Crandall  and  Wing  of  Cor- 
nell University  made  a  careful  and  elaborate  series  of  experiments  on  rollers  i,  2,  3,  and  4 
inches  in  diameter,  on  plates  inches  thick,  using  cast-iron,  wrought-iron,  and  steel  in  both 
rollers  and  plates,  in  all  combinations.  Thirty-three  combinations  of  sizes  and  materials,  and 
in  all  about  two  hundred  observations  of  loads  and  corresponding  areas  of  contact,  were  made. 
These  experiments  have  never  been  published  or  discussed,  but  they  were  loaned  to  the 
authors  of  this  work,  and  they  have  given  them  a  very  thorough  study  and  analysis.  Several 
assumptions  were  made  as  to  the  distribution  of  the  stress,  and  all  abandoned  as  inadequate. 
It  wai,  finally  decided  to  make  a  purely  empirical  analysis  of  the  results  and  obtain  such  equa- 

*  See  a  paper  on  the  Strength  of  Metallic  Rollers,  by  Prof.  Johnson,  in  Journal  Assoc.  Eng.  Soc,  Vol.  IV,  p.  iia 
Also  Prof.  Burr's  Bridge  and  Roof  Trusses. 


276 


MODERN  FRAMED  STRUCTURES. 


tions  as  they  would  give.  The  observations  were  made  by  using  a  coating  of  tallow  to  show 
the  area  of  contact.  It  soon  became  evident  that  the  widths  of  contact  had  usually  been 
taken  too  great,  especially  for  the  lighter  loads  and  for  the  larger  diameters.  When  these 
were  corrected  (the  corrections  being  from  o.Oi  to  0.03  inches,  the  observed  widths  being  re- 
duced by  these  amounts  in  several  instances)  the  analysis  showed  the  following  relations 

Let  /  =  load  in  pounds  per  linear  inch  of  roller ; 
D  =  diameter  of  roller  in  inches  ; 
A  =  area  of  contact  per  linear  inch,  in  square  inches, 

=  width  of  surface  of  contact  in  inches  ; 
/,  —  compressive  stress  in  pounds  per  square  inch  at  centre  of  area  of  contact ; 
=  compressive  stress  at  the  elastic  limit; 
=  working  value  of  the  compressive  stress. 
c,  k,  and  K  =  constants  determined  from  the  observations. 

It  was  found  that 

A=kVpD>  (I) 

where  k  has  different  values  for  the  different  materials. 

B  _  The  distortion  area  ABCD,  Fig.  303,  may  be  con- 

^^"^"'^—^^s.    sidered  as  the  segment  of  a  parabola,  in  which  case  the 

ordinate  BD  is  f  of  the  mean  ordinate.    If  the  stress  be 
/  \         assumed  to  be  proportional  to  the  strains,  or  distortions, 

/  the  maximum  stress  would  also  be  |  of  the  mean  stress. 

°'  Or  we  could  write 

/'=ii  •  w 


From  (i)  and  (2)  we  have 


(3) 


The  first  indentation  of  the  rollers  and  plates  were  not  very  closely  observed  in  these 
experiments,  but  in  a  very  careful  set  of  experiments  made  by  Prof.  A.  Marston,  who,  in  con- 
nection with  Prof.  Crandall,  has  fully  discussed  this  whole  problem,*  the  elastic  limits  of  steel 
rollers  on  steel  plates  were  carefully  determined.  In  these  latter  experiments  eleven  rolleis 
were  employed  from  i  in.  to  16  in.  diameter.  These  results  show  that  the  elastic-limit  load 
with  soft-steel  rollers  on  steel  plates,  per  linear  inch  of  roller,  is 

p.  =  cD  (4) 

where  c  is  a  constant,  and  in  these  experiments  was  found  to  be  880  for  soft-steel  roller  and 
plate.  Hence  p  =  88oZ^  for  soft  steel.  By  combining  eqs.  (3)  and  (4),  we  find  the  elastic  limit 
stress  at  the  centre  of  the  area  of  contact  to  be 


*  See  paper  by  Professors  Crandall  and  Marston  in  Trans.  Am.  Soc.  C.  E.,  Vol.  XXXII.,  p.  99,  and  also  discussion 
of  same,  p.  269. 


DETAILS  OF  CONSTRUCTION. 


277 


f-=v-i  <5) 

Here  both  c  and  k  are  constants  depending  on  the  materials. 

From  both  sets  of  experiments  named  above,  we  can  derive  the  following  approximate 
values  of  k,  c,  and  p^. 


Combination  of  Materials. 

k 

c 

/c 

0.00050 
0.00040 
0.00036 

1460 
g6o 
880 

115,000 
115,000 
124,000 

If  the  working  stress  be  taken  at  about  one  fifth  the  elastic  limit  stress  for  cast-iron,  and 
at  about  one  third  that  stress  for  wrought-iron  and  soft  steel,  we  have  as  the  working  load  per 
linear  inch  of  roller  for  all  combinations  of  cast-iro7i,  wrought-iron,  and  soft  steel,  and  for  all 
diameters  from  i  inch  to  16  inches, 

p  =  300D  (6) 

If        total  load  to  be  carried,  and 

/  =  total  length  of  bearing  rollers,  we  have 

_P  _  P 

p^  ZOoD ^''^ 

In  case  the  load  is  liable  to  be  unequally  distributed  over  the  rollers,  it  may  be  advisable 
to  take  smaller  working  stresses.  As  an  extreme  case,  where  the  load  may  be  concentrated  on 
one  half  the  rollers,  but  using  in  this  case  factors  of  safety  of  3.5  and  of  2  on  the  elastic  limit 
strength,  in  place  of  5  and  3  respectively,  as  before,  we  would  have 

/  =  200Z>  (8) 

or 

P 

 (9) 


256.  A  Good  Design  for  Heavy  Movable  Bearings. — ^Figs.  304  and  305  are  taken 
from  Mr.  Geo.  S.  Morison's  design  for  the  movable  bearings  on  the  middle  pier  of  the 


Fig.  304.  Fig.  305. 


278 


MODERN  FRAMED  STRUCTURES. 


Memphis  Bridge,  there  being  a  fixed  span  of  621  feet  on  one  side  and  a  cantilever  span  of 
7po  feet  on  the  other.  Thus  this  movable  support  sustains  a  length  of  span  of  over  705^^  feet, 
wliich  has  a  dead  load  of  some  5,000,000  pounds  and  a  live  load  of  some  3,000,000  pounds 
more,  or  some  4,000,000  pounds  total  load  on  each  support.  In  order  to  increase  the  bearing 
surface  without  making  the  sole  plate  of  the  shoe  too  great,  the  sides  of  the  cylindrical  rollers 
are  cut  out  and  thus  they  may  be  made  of  large  radius,  as  shown  in  the  accompanying  figures. 
These  rollers  are  15  inches  in  diameter  and  are  spaced  8  inches  apart.  There  are  fifteen  of  them, 
each  112  inches  long,  or  1680  linear  inches  of  roller  under  each  truss  bearing.  This  is  nearly 
2400  pounds  per  linear  inch.  By  our  formula  p—  1200  ^ D  per  lineal  inch.  Since  D  here  is 
15  inches,  our  formula  would  allow  4650  pounds  per  linear  inch,  or  nearly  twice  the  actual  load 
on  these  rollers.  The  distribution  of  the  loads  evenly  over  the  rollers  from  the  pin  is  a  very 
important  matter,  and  the  method  here  adopted  of  using  two  courses  of  I-beams  above  and 
a  course  of  railroad  rails,  with  one  flange  removed  so  as  to  pack  solidly  together,  for  a  base, 
and  below  all  a  high  casting  for  distributing  evenly  upon  the  pier,  is  to  be  commended.  An 
objection  which  can  be  made  to  the  scheme  is  that  it  is  very  expensive. 

257.  Provision  for  Expansion. — All  spans  under  75  feet  usually  rest  on  planed  bed- 
plates and  expand  or  contract  by  sliding  on  the  surface  between  the  sole  and  bed-plate. 
Rollers  are  used  under  the  sole-plates  or  shoes  of  all  spans  over  75  feet.  When  rollers  are 
used  the  span  should  be  supported  on  an  end  pin  so  as  to  insure  the  pressure  coming  on  the 

rollers  centrally.  This  applies  to  plate  and  lattice 
girder  spans  as  well  as  to  pin-connected  truss  spans, 
because  otherwise  the  deflection  of  the  span  would 
necessarily  transfer  the  pressure  to  the  inner  edge  of 
the  sole  plate  and  to  a  few  rollers  at  that  end  of  the 
nest  of  rollers.  Owing  to  the  expense  of  boring  the 
pin-holes  in  a  long  plate  girder,  the  plan  shown  in  Fig. 
306  is  sometimes  used.  In  using  this  plan  care  must 
be  taken  to  make  the  plates  stiff  enough  to  distribute 
the  pressure  over  the  rollers  or  the  masonry.  For  plate 
girders  and  single-web  lattice  girders  (as  distinguished 
from  girders  with  box  chords)  the  plan  Fig.  306  is 
preferable  to  a  shoe  and  pin  owing  to  its  greater  lateral 


o 


o 
o 
o 

o 


o 

o 
o 
o" 
o 


1 


o 


o 


Fig.  306. 
stiffness. 

The  expansion  may  be  provided  for  by  suspending  the  end  of  the  span  on  links.  This 


1-^    rs  Q  


^Cross  Glrdet 

Fig.  307. — Expansion  Joint,  Elevated  Railroad. 


DETAILS  OF  CONSTRUCTION. 


279 


is  sometimes  done  in  elevated-railroad  work,  and  is  always  the  case  with  the  suspended  span 
of  a  cantilever  bridge. 

When  the  plan  Fig.  307  is  used,  the  allowed  bearing  pressure  between  the  pins  and  the 
links  should  be  reduced  from  that  ordinarily  allowed  on  pins  in  order  to  prevent  the  con- 
tinual motion  from  wearing  the  pin. 

In  viaducts  where  the  girders  rest  on  the  caps  of  the  columns  the  girders  expand  by  slid- 
ing on  the  column  caps.  Usually  the  longitudinal  girders  of  elevated-railroad  structures  are 
allowed  to  expand  on  brackets  built  out  from  the  cross  girders. 

The  towers  of  viaducts  are  allowed  to  expand  by  sliding  on  their  bed-plates.  There  are 
two  ways  for  providing  for  this  expansion,  as  shown  in  Fig.  308.    In  plan  (i)  the  base  of  the 


Longitudinal  Transverse 
VIEWS  OF  TOWER 


Fig.  308. 


columns  a  are  fixed  or  anchored  fast  to  the  masonry  while  the  bases  of  columns  b  are 
allowed  to  move  in  the  direction  of  the  arrows.  In  (2)  the  base  of  c  is  fixed  and  the  bases  of 
columns  d  are  allowed  to  expand  in  the  direction  of  the  arrows.  In  each  of  these  cases  the 
bottom  struts  of  the  tower  must  be  made  strong  enough  to  overcome  the  friction  of  the 
column  on  the  bed-plates  considering  the  tower  unloaded. 

In  making  allowance  for  expansion  a  variation  of  temperature  of  150  degrees  is  usually 
provided  for.    The  change  in  length  is  generally  assumed  to  be  one  inch  in  eighty  feet. 

258.  The  Design  of  the  Truss.— For  through  spans  the  Pratt  truss  with  inclined  end 
posts  has  been  found  to  be  the  most  economical  for  spans  under  180  feet ;  from  180  to  225  feet 
the  curved  chord  single  intersection  truss  is  generally  used,  and  above  that  length  the  curved 
chord  single  intersection  truss  with  sub-panels.  For  skew  through  spans  the  straight  chord 
must  be  used  for  all  lengths,  or  a  curved  chord  truss  with  one  inclined  and  one  vertical  end 
post  may  be  used.  In  the  latter  case  the  portal  strut  would  connect  the  vertical  end  post  of 
one  truss  with  the  vertical  hanger  from  the  top  pin  of  the  inclined  post  of  the  other  truss. 
When  inclined  end  posts  are  used  in  a  skew  bridge  the  two  end  posts  that  the  portal  connects 
must  hav9  the  same  inclination  in  order  that  the  portal  may  not  be  a  warped  surface. 

For  deck  spans  the  triangular  truss  will  generally  be  found  to  be  economical.  For  long 
spans  sub  panels  would  be  used.  In  general  deck  spans  are  more  unstable,  unless  made  very 
wide  centre  to  centre  of  trusses,  than  through  spans,  as  the  floor  and  the  train  surface  exposed 
to  the  wind  are  further  from  the  shoe  or  point  of  support.  However,  if  the  deck  span  is  sup- 
ported on  the  end  top  chord  pin  it  is  more  stable  than  a  through  span,  as  the  wind  force  on 
the  bottom  chord  produces  a  negative  moment  to  that  of  the  wind  force  on  the  floor  and  train 
surface.  Deck  ske^v  spans  supported  on  the  end  bottom  chord  pin  are  to  be  avoided  when 
possible  owing  to  difficulty  of  designing  efficient  end  sway  bracing. 

259.  Various  Methods  of  Sub-panel  Trussing. — In  Fig.  309  are  shown  three  methods 
in  use  for  supporting  the  floor-beam  at  the  intermediate  panel  point  in  a  sub-panelled  truss. 


28o 


MODERN  FRAMED  STRUCTURES. 


As  a  question  of  economy  there  is  practically  no  difference  between  them,  (f)  being  a  little 
more  expensive  than  the  others.    Design  ib)  is  to  be  preferred  because  the  primary  concep- 


ia)  (b)  (c) 

Fig.  309. 


tion  of  the  truss  is  to  have  all  diagonals  tension  members  and  hi  all  cases  there  is  no  ambiguity 

of  stresses. 

260.  Stiff  End  Bottom  Chords. — A  great  many  engineers  require  that  the  end  bottom 
chord  members  of  a  through  span  be  made  stiff.  It  has  been  found  that  this  adds  consider- 
ably to  the  rigidity  of  the  span,  but  no  definite  method  of  proportioning  these  members  has 
been  advanced.  The  wind  causes  a  compressive  stress  in  these  members,  but  rarely  enough 
to  overcome  the  tension  from  the  vertical  load.  As  a  matter  of  safety  the  wind  stress  in  the 
bottom  chord  should  always  be  computed  and  the  most  unfavorable  assumptions  possible 
made,  and  if  the  tension  is  overcome,  or  nearly  so,  the  chord  should  be  made  stiff. 

These  stiff  chords  are  made  by  stiffening  the  eyebars  as  shown  in  Fig.  310,  or  by  using  a 


1  m» 

m     1  1 

1                1      1    1  1 

1  mmQmm  1    1  ^  ^    ^   ^   ■ 

C 

c 

r  \ 

^  z 

----- 

00 

1  m 

Uu.a^^M  1          1  ^  9—  W  0  ' 

Fig.  310. 


compound  section  of  four  angles  or  of  two  channels  latticed.  In  either  case  it  must  be  noted 
tiiat  the  section  is  reduced  by  the  rivet-holes,  and  in  proportioning  for  the  tensile  stress  the 
net  area  must  be  used. 

261.  Stiff  Floor-beam  Hangers. — When  a  floor-beam  is  suspended  from  a  pin  some 
distance  above  the  beam  the  hanger  or  tie  supporting  the  beam  is  often  made  stiff.  The 
advantage  of  a  rigid  member  for  such  cases  is  that  greater  stiffness  of  the  entire  structure  is 
obtained.  The  hanger  may  be  made  of  any  of  the  various  forms  mentioned  for  stiff  bottom 
chords. 

Engineers  differ  as  to  the  form  of  the  member  to  use  in  the  case  of  a  stiffened  tie,  some  holding  that  it 
should  simply  be  that  of  a  tension  member  braced  as  shown  in  Fig.  310,  while  others  prefer  that  it  should 
be  made  of  the  ordinary  post  section.  The  form  shown  in  Fig.  310  no  doubt  looks  better,  but  as  the  post 
section  is  the  stiffer  form  and  can  be  made  amply  strong  it  will  make  a  more  rigid  member. 


DE2AILS  OF  CONSTRUCTION. 


281 


\  I  262.  Collision  Struts  are  sometimes  used  to  brace  the  end 
\             post  against  a  horizontal  force,  generally  assumed  to  be  that  of  a 
\          train  off  the  track,  striking  the  end  post,  in  the  plane  of  the  truss. 
 \|/    They  are  redundant  members  of  doubtful  utility  and  are  not  gen- 
erally used.    Fig.  311  shows  the  usual  position  of  this  strut. 
^'                     263.  Sub-struts  are  those  struts  which  are  used  to  stiffen 
compression  members  of  a  truss  by  holding  them  against  flexure.    Thus,  in  Fig.  312  the 


Fig.  312. 


sub-struts  fl:  support  the  top  chord  sections  in  the  middle  and  reduce  their  unsupported  length 
as  a  long  column  one  half.  The  strut  b  similarly  braces  the  main  vertical  posts.  A  saving  in 
material  is  often  accomplished  by  the  use  of  these  struts,  as  the  area  required  in  the  top 
chords  or  posts  to  resist  their  compressive  stress  is  less  owing  to  the  shorter  unsupported 
lengths. 

264.  The  Design  of  the  Floor  System. — The  main  objects  to  be  sought  after  in  the 
design  of  the  floor  system  are  strength  and  stiffness  of  the  floor-beams  and  stringers  and 
rigidity  in  their  attachments  to  the  floor-beams  and  to  the  trusses.  As  the  floor-beam  is 
usually  utilized  as  a  strut  of  the  lateral  system,  it  must  be  located  as  near  as  practicable  to 
the  chords,  which  also  serve  as  chords  for  the  lateral  system,  and  further,  the  beams  must  be 
in  such  a  position  that  the  lateral  rods  maybe  put  in  below  the  cross  ties  and  be  in  the  plane, 
or  nearly  so,  of  the  chords. 

To  secure  beams  and  stringers  of  sufficient  strength  it  is  only  necessary  to  use  sufficiently 
low  unit  stresses  in  proportioning  them.  The  usual  permissible  stresses  in  practice  fulfil  this 
condition.  To  secure  ample  stiffness  the  deflection  of  the  beams  and  stringers  under  load  must 
be  considered.  In  general,  it  may  be  said  that  the  deflection  of  a  beam  varies  directly  as  the  stress 
per  square  inch  on  the  flanges  and  inversely  as  the  depth.*  Hence,  an  increase  in  stiffness  may 
be  obtained  by  increasing  the  depth  of  the  beams  and  stringers  or  by  decreasing  the  permis- 
sible unit  stresses  on  the  flanges.  As  a  reduction  of  the  allowed  unit  stresses  would  result  in 
increasing  the  amount  of  material  necessary  in  the  beams  and  stringers,  it  is  economy  to 
obtain  the  required  stiffness  by  increasing  their  depth.  For  the  usual  unit  stresses  the  stringers 
should  have  a  depth  not  less  than  one  twelfth  of  their  length.  Stringers  having  less  depth 
than  this  would  have  a  perceptible  deflection  which  would  tend  to  loosen  the  rivets  attaching 
the  stringer  to  the  floor-beam. f  The  tendency  at  present  is  to  the  use  of  deep  floor-beams  and 
stringers.  Stringers  having  a  depth  of  one  sixth  to  one  tenth  of  their  length  are  commonly 
used  and  are  generally  the  most  economical  in  material. 

To  secure  the  greatest  amount  of  rigidity  in  the  floor  system  the  stringers  should  be 
riveted  between  the  floor-beams  and  the  beams  riveted  to  a  stiff  member  of  the  truss.  In 
order  to  obtain  a  simple  detail  for  the  floor-beam  in  such  a  case  it  is  necessary  that  the  post 
be  vertical.    This  makes  it  necessary  to  use  redundant  members  in  a  Warren  truss,  and  has 

*  See  column  4  of  tabular  form,  p.  132. 

f  In  many  of  the  old  iron  bridges  now  in  use  the  stringers  are  too  shallow  and  their  end  connections  are  con- 
tinually giving  way  due  to  the  excessive  deflection  bringing  a  tensile  stress  on  the  rivets.  This  weakness  of  the 
riveted  connection  of  the  stringer  to  the  beam  is  often  cited  as  a  fault  of  this  kind  of  connection,  but  it  is  really  due  to  the 
fact  that  the  stringer  is  too  shallow.  Shallow  stringers  sometimes  fail  by  the  splitting  of  the  web  plate  at  the  ends,  a 
failure  which  would  only  occur  in  the  case  of  a  faulty  detail  of  the  ends. 


282 


MODERN  FRAMED  STRUCTURES. 


Fig.  313.  Fig.  314. 


Fig.  315.  "  Fig.  316. 


DETAILS  OF  CONSTRUCTION. 
Fig.  317.  Fig.  318. 


2 


Fig.  319.  Fig.  320. 


I 


284 


MODERN  FRAMED  STRUCTURES. 


I 


led  to  the  adoption  of  the  plate  hanger  (Fig.  317^)  for  the  late  designs  of  the  Pegram  truss. 
In  the  latter  case  the  lateral  connection  is  made  as  shown  in  Fig.  316  or  328. 

The  plate  hanger  detail  is  one  in  which  the  floor-beam  is  riveted  to  a  vertical  plate  which 
has  a  pin-hole  in  its  upper  end  to  receive  the  truss  pin.  The  great  advantage  of  this  detail  is 
that  the  load  from  the  beam  is  applied  centrally  to  each  truss,  thus  insuring  equal  stress  in  the 
tie-rods.  In  some  of  the  older  bridges  in  which  the  floor-beams  are  riveted  to  the  posts  the 
inside  rods  of  each  truss  receive  a  greater  stress  than  the  outer  rods  — a  fact  made  evident  by 
inspection  after  years  of  service. 

Figures  313,  314,  315,  316,  317,  318,  319,  and  320  illustrate  most  of  the  many  designs  for 
the  floor  system  now  in  common  use.  The  designs  shown  in  Figs.  313,  317,  318,  and  320  are 
to  be  preferred  as  they  are  cheaper,  simpler  in  construction,  and  are  for  any  but  the  most 
careful  workmanship  and  best  material  much  safer  than  the  others.  The  bending  of  the 
flange  angles  of  the  floor-beam  or  of  any  girder  should  be  avoided  as  expensive  and  dangerous. 
A  sharp  bend  in  an  angle  iron  is  made  by  cutting  out  a  V-shaped  piece,  then  bending  the 
angle  and  afterwards  welding  the  parts  together.  It  is  very  doubtful  whether  a  perfect  weld 
can  be  made,  and  if  it  is  not  the  value  of  the  angle  is  destroyed. 

The  floor  system  is  subjected  to  severer  stresses  than  the  trusses,  and  the  design  should 
always  be  such  as  will  insure  the  maximum  of  safety. 

It  is  customary  to  use  a  system  of  lateral  bracing  between  the  top  flanges  of  the 
stringers  for  panels  the  length  of  which  is  over  twenty-five  times  the  width  of  the  stringer 
flange.  The  purpose  of  the  bracing  is  to  stiffen  the  stringer  flanges,  and  it  is  usually  made  of 
angle  iron  with  riveted  connections  to  the  stringers. 

265.  The  Attachment  of  the  Lateral  Systems. — Figs.  321,  322,  323,  324,  and  325 
illustrate  the  usual  detail  for  the  connection  of  the  top  lateral  rods  of  through  bridges.  Figs. 
326,  327,  328,  329,  330,  and  331  show  the  usual  details  for  the  attachment  of  the  lower  lateral 
rods  of  through  bridges.  In  all  of  these  connections  the  connection  is  eccentric,  i.e.,  the  rods 
do  not  intersect  on  the  centre  line  of  the  chord  and  they  are  so  far  imperfect,  but  they  are 
illustrations  of  the  various  methods  employed  to  so  connect  the  rods  as  to  produce  as  little 
eccentricity  as  possible.  The  details  Fig.s.  330  and  331  were  formerly  used  quite  extensively, 
but  are  now  seldom  employed.  In  selecting  the  style  of  detail  to  use,  the  designer  must  be 
governed  by  the  special  conditions  of  the  span  he  has  in  hand.  The  connection  must  be  so 
made  that  the  stress  from  the  rod  can  be  transferred  to  the  chord  and  to  the  lateral  strut 
without  overstressing  any  part. 

The  lateral  strut  has  been  omitted  from  the  sketches  of  top  lateral  connections,  as  the  detail 
of  its  attachment  depends  on  the  form  of  the  strut.  The  strut  is  usually  riveted  between  the 
chords.    From  Figs.  313-320  a  fair  idea  of  the  connection  of  the  struts  may  be  obtained. 

For  short  .spans  where  the  wind  or  lateral  stress  is  comparatively  small  any  of  the  designs 
3hown  will  be  satisfactory  if  care  is  taken  to  get  the  laterals  as  nearly  in  the  plane  of  the 
chords  as  practicable. 

For  long  spans  where  the  lateral  stress  is  great  it  becomes  necessary  to  design  a  detail 
which  will  avoid  all  eccentricity  and  the  consequent  overstressing  of  some  member.  No 
satisfactory  detail  for  this  case  has  yet  been  universally  adopted.  The  detail  shown  in  Fig. 
332  was  used  in  some  long  truss  spans  recently  built  over  the  Ohio  River  and  satisfies  all  the 
important  conditions.  The  lower  chord  is  built  in  two  lines  placed  far  enough  apart  to  allow 
the  floor-beam  to  be  riveted  between  them,  the  line  of  the  diagonal  ties  of  the  truss  intersect- 
ing the  centre  line  of  the  post  at  a  point  midway  between  the  two  lines  of  chords.  The 
lateral  rods  also  intersect  at  the  same  point.  The  chord  components  of  the  truss  ties  and  the 
lateral  rods  are  transferred  to  the  chords  by  the  hanger  plate  to  which  the  floor-beam  is 
riveted.  This  hanger  plate  is  made  stiff  enough  to  resist  the  forces  acting  upon  it.  This 
detail  was  designed  by  Mr.  H.  G.  Morse,  President  of  the  Edge  Moor  Bridge  Works. 


288 


MODERN  FRAMED  STRUCTURES. 


266.  Portal  and  Sway  Bracing. — Considerable  attention  has  been  paid  of  late  years  to 
the  design  of  efficient  portal  bracing.  The  fact  that  the  connections  for  the  portal  are  sub- 
jected to  severe  reverse  stresees  and  that  the  stress  in  the  end  post  is  usually  of  considerable 
magnitude  makes  the  problem  very  difificult  and  the  necessity  for  correct  designing  very 
apparent.  The  connection  with  the  end  post  should  be  such  that  it  will  withstand  both 
tension  and  compression,  and  the  end  post  should  receive  its  stress  centrally.  Practically 
these  conditions  cannot  readily  be  fulfilled,  as  it  is  probably  impossible  to  make  a  central 
attachment  to  the  end  post.  The  usual  plan  is  to  rivet  the  portal  strut  between  the  end 
posts  and  rely  on  the  rivets  in  the  connection  for  the  transfer  of  the  stress.  For  the  shorter 
spans  this  has  proved  by  experience  to  be  sufficient,  but  nevertheless  there  are  parts  which 
are  stressed  beyond  what  would  ordinarily  be  allowed.  For  long  spans  the  portal  strut  is 
now  usually  made  a  box  girder  with  the  webs  in  the  planes  of  the  top  plates  and  of  the 
under  sides  or  tie  plates  of  the  two  end  posts.  The  connecting  plates  extend  across  the  entire 
width  of  the  end  post  in  both  planes.  This  avoids  direct  tension  on  rivets  and  is  as  nearly 
central  as  it  is  possible  to  make  the  joint. 

Skew  portals  increase  the  difficulties  in  the  design,  as  they  bring  in  a  very  serious 
question  as  to  the  manufacture.  They  are  difficult  to  make  fit,  requiring  very  careful  and 
accurate  work,  and,  as  in  all  similar  cases  requiring  close  work,  the  designer  is  necessarily 
limited  in  the  style  of  his  detail  or  connection.  Skew  bridges  should  always  be  designed  with 
parallel  end  posts. 

The  portal  if  necessarily  shallow  is  composed  of  one  strut  the  cross-section  of  which  is 
similar  to  a  plate  or  single-web  lattice  girder.  If  the  depth  of  truss  admits  of  it,  a  strut  at  the 
top  of  the  end  post  and  one  as  far  down  as  the  required  headroom  will  allow,  with  diagonal 
rods  or  braces  between,  is  used.  Figs.  333,  334,  and  335  illustrate  the  common  forms  used  for 
portals  and  the  stresses  in  the  various  parts.    Fig.  335  would  be  a  portal  composed  of  one 


strut  with  knee  braces,  and  is  the  common  form  for  short  spans.  The  stresses  *  are  given  in 
the  most  convenient  terms  to  use. 

For  through  bridges  in  which  the  height  of  truss  will  admit  of  it  intermediate  sway  brae 
ing  is  put  in  between  the  posts  at  each  panel.    This  sway  bracing  usually  consists  of  a  strut 


*  To  Mr.  W.  F.  Gronau  of  Pittsburg,  Pa.,  belongs  the  credit  of  having  computed  the  stresses  using  the  easily  de 
lermined  dimensions  given  in  the  formulae. 


DETAILS  OF  CONSTRUCTION. 


289 


connecting  the  two  opposite  posts  of  the  two  trusses,  and  diagonal  rods  in  the  plane  of  and 
connecting  the  top  strut  and  this  sway  strut.  The  rods  are  usually  of  the  minimum  size 
used,  as  their  function  is  merely  to  prevent  vibration. 

Where  the  height  of  truss  will  not  permit  of  sway  bracing  like  the  above,  knee  braces, 
see  Fig.  314,  are  often  used. 

For  deck  bridges  intermediate  sway  bracing,  see  Figs.  318-320,  is  put  in  at  each  panel. 
The  rods  in  this  case  also  need  only  be  of  the  smallest  size  used  unless  it  be  a  special  case 
where  extra  stiffness  is  required.  This  would  be  in  the  case  of  a  curve  on  the  bridge  or  when 
the  rods  are  unusually  long. 

Sway  bracing  should  be  used  at  the  ends  of  all  deck  spans  of  sufficient  strength  to 
transfer  the  wind  force  from  the  top  lateral  system  to  the  shoe.  This  end  bracing  will  be 
the  most  efficient  if  put  in  the  plane  of  the  end  posts  which  carry  the  load  to  the  shoe. 
These  posts,  having  the  largest  sectional  area,  are  always  stressed  and  are  consequently  stiff. 

The  present  tendency  seems  to  be  toward  the  use  of  angle-iron  diagonals  with  riveted 
connections  for  sway  bracing  instead  of  the  adjustable  rod  with  the  pin  connection.  It  adds 
very  little  if  anything  to  the  cost  to  use  the  angle  iron  or  "stiff"  bracing,  as  it  is  termed, 
instead  of  rods  and  adds  very  materially  to  the  rigidity  of  the  structure. 

267.  Top  Chord  Joints. ^ — The  splices  or  joints  of  the  top  chord  are  made  as  shown  in 
Figs-  336,  337,  338,  339,  and  340,  the  most  common  ones  being  that  shown  by  Fig.  336  for  an 
intermediate  top  chord  joint  and  that  shown  by  Fig.  340  for  the  joint  at  the  top  of  the 
inclined  end  post.  For  the  intermediate  joint  the  two  chord  sections  are  planed  off  to  a  true 
surface  and  the  bearing  of  the  sections  on  each  other  is  relied  on  to  transfer  the  stress.  Side 
and  top  splice  plates  are  riveted  to  each  section  to  hold  them  in  position.  The  field  rivets  in 
these  splices  must  be  so  located  that  they  will  be  accessible  for  driving  after  the  chords  posts 
and  ties  are  in  place.  The  splice  is  usually  located  on  the  side  of  the  pin  furthest  from  the 
centre  of  the  span.  The  hip  joint  or  that  at  the  top  of  the  end  post  (see  Fig.  340)  is  made  in 
a  different  manner;  the  two  sections  do  not  bear  on  each  other,  but  an  opening  of  from  one 
quarter  to  three  eigliths  of  an  inch  is  left  between  them,  and  the  bearing  plates  on  the  pin  for 
each  section  are  made  thick  enough  to  transfer  the  stress.  This  style  of  joint  is  often  used 
for  those  intermediate  top  chord  joints  in  curved  chord  bridges  where  the  two  sections  of 
chord  make  an  angle  with  each  other,  instead  of  the  joint  shown  in  Fig.  339. 

268.  The  Shoe. — The  reason  for  the  use  of  a  shoe  at  the  end  of  a  span  is  to  insure  a 
uniform  pressure  on  the  rollers  or  to  secure  an  evenly  distributed  pressure  on  the  masonry. 
The  usual  limits  for  the  bearing  of  the  shoe  on  the  masonry  are  from  200  to  300  pounds  per 
square  inch.  In  order  that  the  pressure  may  be  uniform  on  the  rollers  or  the  masonry  it  is 
necessary  that  the  shoe  be  stiff  enough  to  so  distribute  it.  This  requires  that  the  ribs  of  the 
shoe,  if  they  are  made  of  the  thickness  required  for  the  bearing  of  the  end  pin,  must  also  be  made 
deep  enough  to  give  the  required  stiffness.  Very  deep  shoes  are  however  to  be  avoided  or 
else  they  must  be  given  ample  lateral  stiffness,  as  the  ribs  of  the  shoe  are  relied  on  to  transfer 
all  the  wind  force  from  both  the  top  and  bottom  lateral  systems  to  the  masonry. 

When  two  shoes  rests  on  one  pier  it  is  always  better  to  have  them  rest  on  one  continuous 
bed-plate  under  both  shoes.  This  plate  acts  as  a  tie  to  bind  the  pier  together  and  to  prevent 
the  friction  of  the  shoes  from  cracking  the  masonry. 

269.  The  Packing  of  Joints. — The  members  connecting  on  a  pin  should  always  be  so 
arranged  as  to  produce  as  small  a  bending  moment  as  practicable,  and  they  should  always  be 
arranged  symmetrically.  Ample  clearances  for  inaccuracies  such  as  are  liable  to  occur  in 
manufacture  should  be  allowed.  The  pieces  should  be  so  placed  that  no  cutting,  or  as  little 
as  possible,  of  the  flanges  of  compression  members  is  necessary  for  fits.  Always  keep  in 
mind  that  the  lateral  rods  must  be  located  as  nearly  as  possible  in  the  plane  of  the  chords. 
Eyebars  should  never  be  bent  in  order  to  pack  nicely  on  the  pins  they  connect,  but  it  should 


290 


MODERN  FRAMED  STRUCTURES. 


■ — ■  1        r\  r\ 

•  • 

•  • 

•  • 

1           b  \ 

|i  1  J  [ 

 - — -i-J 

^  ^   ^  /- 

•  • 

•  • 

•  • 

1  !'^«t( 

1  \/  /  \^ 

0  hh 

000  0/ 

5-sio  i 

 ^\  h 

0\)  0 

/  / 

4/ 

fr 

Li 

-  r^^r^i 
1  \ 

!  ^ 

1  1 
1  1 
1  1 
1 

1  i 

\ 

\ 

> 

Fig.  337. 


Fig.  33S.  Fig.  339, 


r\  r-\    r~\  r\ 

o 
o 
o 


o\o 
oib 


o 
o 
o 


Fig.  340. 


DETAILS  OF  CONSTRUCTION. 


be  an  invariable  rule  to  have  them  as  nearly  parallel  to  the  plane  of  the  truss  as  possible.  De- 
tails  which  require  accurate  work  or  fitting  in  the  field  should  be  avoided,  as  the  chances  are 
that  in  the  hurry  to  "  swing"  the  span  it  will  be  neglected.  Details  should  be  used  which  will 
facilitate  the  work  of  erecting  if  at  no  sacrifice  of  strength. 

270.  Camber. — Bridges  are  so  constructed  that  they  will  when  loaded  to  their  capacity 
take  the  form  which  was  assumed  in  the  calculation  of  the  stresses.  This  is  accomplished  by 
curving  the  trusses  upward,  i.e.,  giving  them  a  camber.  This  is  done  by  increasing  the  length 
of  the  top  chord,  decreasing  the  length  of  the  lower  chord,  and  making  the  corresponding 
necessary  changes  in  the  length  of  the  diagonals.  Formerly  the  rule  was  to  give  all  trusses  a 
camber  or  raise  the  centre  of  the  span  one  twelve-hundredth  of  the  length  of  the  span.  This 
rule  has  now  been  very  generally  superseded  by  one  which  more  nearly  satisfies  the  theoretical 
conditions,  and  that  is  to  make  the  top  chord  one  eighth  of  an  inch  longer  than  the  bottom 
chord  for  every  ten  feet  in  length  of  span.  The  old  method  would  make  the  camber  the 
same  for  all  depths  if  the  span  length  were  constant,  while  the  new  recognizes  almost  per- 
fectly the  fact  that  the  deeper  the  truss  the  less  the  deflection  under  load  will  be  and  hence 
the  less  the  camber  should  be.  The  following  formulae  will  be  found  to  be  serviceable  in 
finding  the  camber  when  the  increase  in  length  of  the  top  chord  over  the  bottom  chord  is 
known,  or  to  find  the  necessary  increase  in  length  of  top  chord  when  the  camber  is  known. 

Let  c  =  camber  in  inches  ; 

i  —  increase  in  length  of  top  chord  over  the  bottom  chord  in  inches ; 
h  =  height  of  truss  in  feet  ; 
/  =  length  of  span  in  feet. 

Then  ^  ~  8^'      ^         *  ~  ~7~* 

In  some  bridge  works  it  is  customary  to  increase  or  diminish  all  truss  members  by  the 
amount  of  their  estimated  distortion  under  their  maximum  loading.  This  gives  a  camber 
equal  to  the  deflection  as  computed  by  the  method  given  in  Chapter  XV. 

271.  Sizes  of  Lattice  Bars. — The  following  table  gives  the  ordinary  sizes  of  lattice  bars 
used  on  compression  members  of  railroad  bridges.  By  the  depth  of  the  member  is  meant  the 
size  of  the  channels,  rolled  or  compounded  of  plate  and  angles,  which  are  used  in  the 
member. 


Depth. 

Lattice. 

Depth. 

Lattice. 

in. 
7 

8 

9 
10 

12 

If  Xi 

2  X^ 
2  Xt\ 
2  Xf 

2iX| 

in. 
14 
15 
16 

18 

over  18 

in. 
2iX| 
2*  X  1 
2iX| 

(4   XI  single 
/  2*  X  f  double 
(4   X  TS  single 

2i  X  1  double 
(3   X  2  X  i  L  single 

292  '  MODERN  FRAMED  STRUCTURES. 


CHAPTER  XIX. 
THE  PLATE  GIRDER. 
THEORETICAL  TREATMENT. 

272.  The  Moments  and  Shears  in  a  plate  girder  at  any  section  are  found  from  the  outer 
forces  in  the  same  manner  as  for  a  section  of  a  truss,  and  as  explained  in  Chapter  II.  The 
relations  of  moments  and  shears  in  a  solid  beam  and  also  in  an  I  beam  or  in  a  plate  girder  are 
explained  in  Chapter  VIII.  It  is  there  shown  that  the  vertical  shearing  stress  in  the  web  of  a 
plate  girder  is  nearly  uniform  throughout  its  depth,  at  any  vertical  section,  and  in  the  follow- 
ing analysis  it  will  be  assumed  to  be  uniformly  distributed.  The  bending  moment  comes 
from  the  shear  acting  with  lever  arms  measured  longitudinally  along  the  girder.  The 
resisting  moment  is  developed  first  in  the  web  and  is  transferred  to  the  flanges  through  the 
rivets  as  the  web  distorts  under  the  moment  which  is  developed  in  it.  The  moment,  there- 
fore, is  primarly  in  the  web,  and  the  amount  of  bending  moment  which  is  resisted  by  the  web 
is  necessarily  such  a  proportion  of  the  bending  moment  at  a  given  vertical  section  as  the 
moment  of  resistance  of  the  web  is  to  the  total  moment  of  resistance  of  the  girder  at  that 
section.  Since  the  web  and  flanges  distort  together  as  a  solid  beam,  there  is  no  reason  why 
the  web  should  not  be  assumed  to  resist  its  due  proportion  of  the  bending  moment.  There 
can  be  no  question  but  that  it  does  actually  perform  this  service. 

Let  F  =  area  of  one  flange  section,  not  counting  the  included  portion  of  the  web; 
h  —  height  of  girder  between  centres  of  gravity  of  flanges  ; 
t  =  thickness  of  plate  in  web  ; 
f  —  stress  allowed  per  square  inch  in  flanges. 

Then  the  total  moment  of  resistance  of  the  girder  is 

M,  =  Ffh-\-^-^  (I) 

Calling  th  =  A  ■=  area  of  web,  we  have,  as  the  moment  of  resistance  of  the  web, 

fth^  _Afh  A„ 

Hence 

M.  =  [f  -^^fk   (2) 

This  shows  that  the  influence  of  the  web  in  resisting  bending  moment  is  fully  provided 
for  when  one  sixth  of  its  area  is  added  to  each  flange  area,  as  indicated  in  equation  (2). 

Where  there  is  a  vertical  line  of  rivet-holes  in  the  web  plate  the  influence  which  these 
holes  have  in  reducing  the  moment  of  resistance  of  the  web  must  be  taken  into  account.  The 
assumption  that  the  net  moment  of  resistance  of  the  web  is  equivalent  to  a  flange  area  of  one 
eighth  of  the  gross  area  of  the  web  plate  is  not  far  in  error  in  any  practical  case.    It  must 


THE  PLATE  GIRDER. 


293 


also  be  borne  in  mind  that  the  web  has  the  same  effect  on  both  tension  and  compression 
flanges,  and  that  it  is  not  correct  to  use  one  sixth  of  the  gross  area  of  web  for  the  web 
equivalent  in  the  compression  flange  and  to  use  one  sixth  of  the  net  area  of  the  web  as  the 
corresponding  equivalent  for  the  tension  flange,  as  is  sometimes  done.  The  web  resists  a  cer- 
tain amount  of  the  bending  moment,  and  the  flanges  must  be  proportioned  for  the  remainden 
Making  this  change  in  equation,  (i)  and  (2),  we  have,  for  practice, 

M,  =  Ffh-{-^  (lA) 

and 

^/o  =  (^+ ^)//^-    .    .    '  (2A) 


000 


000 


000 


3 


-G— 


273.  The  Web  of  a  plate  girder,  or  of  a  floor-beam  or  stringer,  is  made  of  a  single 

plate  if  possible.  In  general,  web  plates  are 
limited  by  the  conditions  of  manufacture  to  a 
net  weight  of  about  1600  pounds  for  standard 
prices.  If  more  than  one  plate  is  required,  it  is 
customary  to  make  up  the  web  symmetrically  as 
to  the  splices  in  it. 

In  light  work,  as  for  highways,  a  minimum 
thickness  of  one  fourth  inch  may  be  used  up  to  a 
width  of  5  or  6  feet.  Such  plates  are  apt  to  be 
more  or  less  buckled,  however.  For  railway  work 
a  minimum  thickness  of  three  eighths  of  an  inch 
should  be  used  for  all  depths  ;  this  thickness  to 
be  increased  if  necessary  to  give  sufificient  bear- 
ing area  on  the  rivets  at  the  flanges. 

274.  Plain  Web  Splices. — The  web  carries 
all  the  shear  (or  nearly  all,  and  is  assumed  to  carry 
all,  see  Art.  130)  and  its  due  proportion  of  the 
bending  moment.*  When  it  has  to  be  spliced, 
the  shearing  and  bending  stresses  at  that  section 
must  be  provided  for  by  double  splice  plates 
with  a  sufificient  number  of  rivets.  Witli  a  three- 
eighths  web,  five-sixteenths  spHce  plates  would  be  used, 
to  use  in  such  a  splice, 


000 


Fig.  341, 

To  find  the  proper  number  of  rivets 


Let  5  = 

shear  on  the  section  ; 

moment  at  the  section  ; 

F  = 

area  at  one  flange  ; 

th  =r 

area  of  the  web  ; 

/  = 

fibre  stress  per  square  inch  in 

the  flange  at  that  section  ; 

r  — 

resistance  of  one  rivet  ; 

2n  = 

number  of  rivets  on  one  side 

of  splice ; 

Then  we  have,  from  eq.  (2A), 


(3) 


*  When  the  flanges  are  proportioned  for  carrying  all  the  bending  moment,  the  web-splice  may  be  proportioned  for 
carrying  the  shear  only. 


294 


MODERN  FRAMED  STRUCTURES. 


This  is  the  flange  stress  at  that  section.  Having  this,  we  find  the  total  moment  carried  by 
the  web  from  the  usual  formula, 

M^-—^  (4) 


The  splice  must  provide,  therefore,  for  the  shear  5  and  the  moment  M^.  Since  the 
shear  is  supposed  to  be  uniformly  distributed  over  the  web  (see  Art.  272),  the  shearing 
stress  on  each  rivet  is 

''^^2^  (5) 


For  resisting  the  bending  moment  on  the  web  the  rivets  are  not  equally  stressed.  (  Since 
the  bending  stresses  are  zero  at  the  neutral  axis  and  increase  uniformly  to  the  maximum  value 
at  the  extreme  rivet,  the  moment  stress  in  each  rivet  will  be  as  its  distance  from  the 
neutral  axis.  Its  arm  is  also  as  this  distance  ;  hence  the  moment  of  resistance  of  the  rivets 
are  as  the  squares  of  their  distances  from  the  neutral  axis.^ 

If  ,  d.^,  d^,  etc.,  are  the  distances  of  the  rivets  from  the  neutral  axis,  then  the  moments 
of  resistance  of  the  several  rivets  will  be  ad^,  ad^,  ad^'',  etc.,  on  one  side,  and  ad',  ad^ ,  ad^, 
etc.,  on  the  other.  If  the  rivets  on  one  side  of  the  joint  be  taken  in  pairs,  symmetrical  about 
the  neutral  axis,  the  moments  of  resistance  of  the  several  pairs  are  2ad^,  2ad^,  2ad^,  etc., 
where  a  is  the  resistance  of  one  rivet  at  a  unit's  distance  from  the  neutral  axis.  The  sum 
of  the  moments  of  the  several  pairs  of  rivets  must  equal  the  total  moment  of  resistance  of  the 
web  at  this  section,  as  given  by  eq.  (4).    Therefore  we  have 


or 




2{d:+d:+d: . .  .  +  d:)' 


a  = 


(6) 
(7) 


Since  a  is  the  rate  of  increase  in  the  stress  on  the  rivets  out  from  the  neutral  axis,  it  is 
equal  to  the  bending  stress  on  the  extreme  rivet  divided  by  its  distance  out,  d„.  The  allow- 
able stress  from  cross-bending  on  this  rivet  is  equal  to  V r'  —  r^—'i'mi  since  the  shearing 
stress,  ,  and  the  stress  from  bending,  r,„ ,  act  at  right  angles  to  each  other  and  both  combine 
to  produce  the  allowable  stress  r.    We  have,  therefore, 

-  ^;  -  2{d:  +   +    . .  +  d:y  •  •  (8) 

or 

^  ^   .  X 

2r,„ 

But  .—  (i'  +  2'  +  3'  +  etc.)(pitch)',  and  the  sum  of  the  squares  of  the  serial  numbers 

«(« -I-  l)(2« -(-  1)  ^, 

I,  2,  3,  .  .  .  «  is  equal  to   g  .    Therefore  we  may  write 


6^{d')  =  7i{n  +1  )(2«  +  iXpitch)'  =  ^Mj^  *.  (10) 


d„ 

But  pitch  =  — ;  hence  we  may  write 


(«  +  i)(2«  +  I)  _  3^ 


  —7-,   .  (II) 


THE  PLATE  GIRDER. 


295 


where    n  =  one  half  the  number  of  rivets  in  a  web  splice  on  one  side  of  the  joint : 
My,  —  the  moment  carried  by  the  web, 

=  — g--,  where/,  is  given  by  eq.  (3), 

=  distance  out  from  neutral  axis  to  extreme  rivet  in  splice  ; 


r„  =  working  resistance  of   extreme   rivet   to  bending   stress  =  V r''  —  Ts  ,  where 

S 

r  =  total  resistance  of   rivet  and  r,  =  shearing  stress  on  rivet  = 


2« 


Equation  (ii)  would  be  solved  by  trial.    The  splice  plate  may  have  to  be  made  so  wide 
as  to  admit  of  two  or  more  rows  of  rivets,  when  they  should  be  staggered. 
From  these  three  equations, 


5 


~  271  • 


(«+l)(2«  +  l)  iM^ 


we  may  find  the  three  unknown  quantities  r,,  r,„,  and  n. 
After  eliminating     and  r„, ,  we  obtain 

Cin+.t2n^.)^  ^^^^eM^]  

l_  n  d„  I 

This  equation  can  best  be  solved  by  trial.  - — 

Example. — Design  a  splice  joint  for  the  web  of  a  50-foot  plate  girder  railway  bridge,  60  inches  deep, 
the  splice  occurring  12}  feet  from  the  end.  Let  the  flanges  at  this  section  contain  13.5  square  inches  each 
(=  f)  and  the  web  22.5  square  inches.  The  live  load  bending  moment  here  from  a  100-ton  engine 
would  be  about  530,000  foot-pounds  and  from  dead  load  235,000  foot-pounds  or  a  total  moment  of  9,180,000 
inch-pounds  =  Afi.    The  total  live  and  dead  load  shear  would  be  53.500  lbs.  =  S. 

From  eq.  (3) .we  have,  as  the  unit  chord  stress, 

M,                 9,180,000  ,,  .  , 

/.    -  —— -  =  -.  — - —  =  9550  lbs.  per  square  inch. 

From  eq.  (4)  we  have 

9550  X  22.5  X  60 
M.U,  =  — g —  =  g  =  1,612,000  inch-pounds. 

This  is  the  total  bending  moment  to  be  resisted  by  the  web  plate  splice. 
From  eq.  (12)  we  have,  for  r  =  4000  lbs.,  5  =  53,500,  and  t/„  =  24, 

(n  +  i)(2n+i)     6x1,612,000 

 i   V  64,000,000;/''  —  2,860,000,000  = 


24 


or 


or 


 li^  ^"  -  45  =  403.000, 


(«  +  l)(2«  +  I) 


V"'  -  45  =  50, 


from  which  we  find,  by  trial, 

n  =  25. 

That  is  to  say,  if  both  the  shear  and  the  moment  credited  to  the  web  plate  at  this  section  are  to  be  trans- 
mitted through  the  rivets  of  the  splice  plates,  with  a  maximum  stress  of  4000  lbs.  on  one  rivet,  it  will  require 
50  rivets  on  each  side  of  the  joint,  or  100  rivets  all  told,  in  this  splice  plate.    This  would  require  three  rows 


296 


MODERN  FRAMED  STRUCTURES. 


of  |-in.  rivets  each  side  of  the  splice,  16  rivets  in  the  two  outside  rows  and  17  rivets  in  the  middle  row,  all 
with  3-inch  pitch. 

It  is  evident  at  once  that  such  a  splice  is  heavy  and  expensive,  and.  that  some  other  means  should  be 
sought  for  transmitting  the  bending  stresses  across  the  joint.  The  rivets  near  the  neutral  axis  are  of  no 
appreciable  assistance  for  this  purpose. 

Such  a  splice  as  that  shown  in  Fig.  341  is  therefore  wholly  incompetent  to  transmit  the  web  bending 
stresses.  In  such  a  case  these  stresses  pass  through  both  the  splice  plate  and  the  flange,  thus  producing 
very  much  larger  rivet  and  flange  stresses  than  they  were  designed  to  carry.  When  such  splices  as  shown 
in  this  figure  are  used  the  flanges  should  be  designed  for  carrying  all  the  bending  moment. 

275.  An  Efficient  Web  Splice. — If  the  rivets  in  the  flange  angles  are  already  stressed 
up  to  their  working  limits  to  transmit  the  flange  stress,  the  bending  stress  in  the  w^eb  should 
be  carried  directly  across  the  joint  through  splice  plates  and  into  the  web  again  on  the  other 
side,  without  going  through  the  flange  angles,  plates,  or  rivets.  These  direct  stresses  (com- 
pression at  top  and  tension  at  bottom)  are  most  efificiently  transmitted  through  long  splice- 
plates  placed  just  inside  the  angles,  as  shown  at  AB  and  A' B'  in  Fig.  342.  Let  the  distance 
between  centres  of  these  plates  be  d„,.  Then  we  have,  as  the  total  stress  transmitted  through 
one  pair  of  plates,  ' 


Stress  in  splice  plates  AB 


and  also 


Number  of  rivets  in  one  end  of  splice  AB  —  —j- 

f  CI  J., 


(13) 


(14) 


From  these  two  equations  the  net  areas  of  the  plates  and  the  numbers  of  rivets  required  to 
transmit  the  bending  stresses  are  found. 


O  II  o      o  o 


0 

0 

0 

0 

0 

0 

0  0 

0 

0  0 

0 

0 

0 

0 

0 

0 

0 

0 

0     0  1 

0  0 

0 

A' 

0 

0  0 

000 

0 

0      0  1 

0  0 

b 

O       O       O    li  o 


0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

B 


\ 


B' 


Fig.  342. 

Thus  with  the  previous  example  we  have  from  eq.  (4), 

Mw  —  1,612,000  inch-pounds. 
r  =  4000  lbs.  per  rivet. 


THE  PLATE  GIRDER. 


297 


It  the  vertical  leg  of  the  flange  angle  be  4  inches,  and  the  splice  plates  AB  be  8  inches  wiae,  the  distance 
between  centres,  d,n  ,  will  be  60  —  16  =  44  inches.  The  stress  in  the  plate  will  be,  therefore,  37.000  lbs.,  re- 
quiring 9  rivets  on  each  end  of  each  pair  of  plates,  as  shown  in  Fig.  342. 

53,500  .  ... 

The  shear  will  be  carried  by  the  splice  plate  CD  with    =  13  rivets  on  each  side. 

'  4000 

This  joint  has  now  62  rivets  in  place  of  100  required  for  a  uniform  plate  and  rivet  distribution  as 
computed  in  the  previous  article. 

The  splice  plates  AB  should  not  extend  over  the  vertical  legs  of  the  angles,  since  this 
would  give  double  duty  to  the  rivets  in  the  angle.  With  a  web  spliced  as  here  designed  there 
is  no  objection  to  designing  the  girder  flanges  on  the  common  assumption  that  one  eighth  of 
the  area  of  the  web  is  added  to  each  flange  area  to  resist  bending  moment. 

276.  Distribution  of  Concentrated  Loads  Over  the  Web.— Since  the  flange  stresses 
arc  first  developed  in  the  web  from  the  shear,  any  external  force,  whether  coming  on  the  top 
of  the  girder  as  a  load,  or  on  the  bottom  as  the  end  support,  must  be  distributed  through  the 
web  by  means  of  vertical  stiffeners,  which  are  usually  angle  irons.  ^ The  number  of  rivets  in 
these  is  equal  to  the  total  external  load  divided  by  the  resistance  of  one  rivet^  These 


^0 

000 

0 

0 

0  i 

1 — 1 

|o 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

1 

0 

0 

0 

0 

0 

0 

i 

0 

1 

0 

0 

i 
1 

! 

0 

1 

( 

j 

1 
1 

0 

0 

000 

0 

0 

0  1 

io 

0 

0 

0 

0 

0  ' 

1 — 

:0 

1 

0 

0 

0 

0 

0 

Fig.  343- 


stiffeners  should  extend  over  the  vertical  legs  of  the  flange  angles  and  abut  against  the 
inside  surfaces  of  the  outer  legs  of  these  angles,  being  either  bent  over  the  vertical  leg  or 
supported  by  a  filler-plate  of  the  same  thickness  as  the  flange  angle. 

277.  Web  Stiffeners. — To  assist  the  web  plate  to  resist  the  compressive  stresses  acting 
at  45°  with  the  axis  of  the  girder,  as  explained  in  Art.  130,  and  to  insure  against  the  buckling 
of  the  web  under  this  stress,  angle  irons  are  riveted  to  the  web,  usually  in  a  vertical  position. 
They  would  be  much  more  efficient  if  put  in  an  inclined  position,  extending  outward  and 
downwards  towards  the  abutment.  The  vertical  stiffeners  are  sufficient  to  do  the  work  if 
placed  somewhat  less  than  the  depth  of  the  girder  apart.  If  placed  farther  apart  than  this 
they  do  very  little  good,  since  the  buckling  tendency  can  have  a  free  development  between 
the  adjacent  stiffeners.  It  is  not  possible  to  rationally  design  these  stiffeners.  The  equal  ten- 
sile stresses  at  right  angles  to  the  compressive  stresses  in  the  web  tend  to  prevent  the  buck- 
ling and  cause  it  to  develop  in  short  curves,  if  it  occurs  at  all.  Almost  any  angles,  however 
light,  placed  as  here  described  (shown  in  dotted  lines  in  Fig.  343),  will  serve  to  prevent  buck- 
ling in  a  f-inch  web  of  the  ordinary  depths.  No  column  formula  can  be  made  to  apply  to  the 
web  of  a  plate  girder. 

278.  Flange  Areas. — If  the  web  is  designed  to  carry  its  due  proportion  of  the  bending 


298 


MODERN  FRAMED  STRUCTURES. 


moment,  as  given  by  eq.  (4),  then  the  remainder  only  is  to  be  carried  by  the  flanges.  Thus 
from  ea.  (3)  we  obtain,  as  the  moment  carried  by  the  flange, 


The  flange  area  is,  therefore, 


ftW 

Mf  -  f,Fh  =  M,  —  ^  (15) 

 (16) 


This  area  is  made  up  of  two  angles  and  one  or  more  cover  plates.  The  area  of  the  web 
included  between  the  angles  is  considered  a  part  of  the  web  and  not  a  part  of  the  flange. 
The  rivet-holes  are  to  be  deducted  from  the  total  area  of  the  tension  flange.  Unequal-legged 
angles  are  commonly  used  so  as  to  throw  the  centre  of  gravity  of  the  angles  as  high  as  possible 
and  also  to  make  the  flange  wide,  and  therefore  rigid  against  lateral  deflection  or  buckling. 

279.  Distribution  of  Rivets  in  the  Flanges.— The  pitch  of  the  rivets  in  the  flange 
angles  and  web  plate  is  usually  determined  by  computing  total  flange  stress  at  intervals  along 
the  beam  and  dividing  the  stress  increments  in  the  flanges  by  the  resistance  of  one  rivet. 
The  rivets  being  in  double  shear  and  the  web  plate  thin,  this  resistance  is  always  determined 
by  the  bearing  area  on  the  web  plate.  A  better  solution  is  as  follows,  by  which  the  proper 
pitch  to  give  to  these  rivets  at  any  section  is  at  once  found. 

Let  Fig.  344  represent  any  portion  of  a  plate  girder,  the 
1  shear  on  the  section  through  AB  being  5.    Taking  moments 
about  D,  and  assuming  for  the  present  that  all  the  moment 
is  resisted  by  the  flanges,  we  have 


B  c 


A  D 


Sp  =  rh,  or 


rh 


(17) 


Fig.  344. 


where  S  =  shear  on  the  section  ; 

p  —  pitch  of  rivets  in  flange  angles  ; 
r  ■=  resistance  of  one  rivet ; 

h  —  distance  between  rivet  lines  in  top  and  bottom 
flanges. 

If  the  flanges  are  designed  to  carry  all  the  bending  mo- 
ment, this  equation  would  give  the  pitch  of  the  rivets  at 
any  section.  From  eq.  (2),  however,  we  see  that  one  eighth  of  the  area  of  the  web  is  equally 
effective  with  that  of  the  flange  in  resisting  the  moment.  Therefore  the  portion  of  the 
moment  developed  at  this  section  arising  between  rivets,  which  is  equal  to  Sp,  is  only  partly 
resisted  by  the  flange  rivet.    The  pottion  going  to  the  flange  is  always  equal  to 


AMF 


SpF 


—  rh, 


(18) 


and  therefore 


F 


8 

A_ 

8  rh 
5' 


(19) 


where  I^-  area  of  one  flange  ; 
area  of  web. 


THE  PLATE  GIRDER, 


1 

I 

299 


PRACTICAL  DESIGNING. 


280.  The  Detail  Design  of  a  Deck  Plate-girder  Span. — Let  the  span  be  assumed 
to  be  47  feet  6  inches  in  the  clear.  The  distance  from  the  base  of  rail  to  masonry  to  be 
made  to  suit  the  girders  as  designed.  The  live  load,  or  capacity,  to  be  Cooper's  "  Class 
Extra  Heavy  A"  loading.  (A  table  of  the  bending  moments  and  shears  for  this  load  is  given 
on  page  329).  The  floor  is  assumed  to  weigh  400  lbs.  per  linear  foot  of  track.  The  allowed 
stress  per  square  inch  on  the  net  area  of  lower  flanges  to  be  15,000  lbs.  for  the  dead  load 
stress  and  7500  lbs.  for  the  live  load  stress.  The  allowed  shearing  stress  on  rivets  to  be 
6000  lbs.  per  square  inch  and  the  allowed  bearing  pressure  of  rivets  on  the  web  plate  to  be 
12,000  lbs.  per  square  inch.  The  top  flanges  to^  be  of  the  same  gross  area  as  the  bottom 
.flange,  and  at  least  one-half  the  area  of  the  flanges  to  be  in  the  angles.  The  pressure  on  the 
masonry  to  be  limited  to  250  lbs.  per  square  inch.  "  ' 

Assuming  a  length  of  fifty  feet  centre  to  centre  of  end  bearings,  and  that  the  iron  weight 
is  given  by  the  formula  9  X  /  +  i  io>  we  get  an  end  reaction  or  total  pressure  on  the  masonry 
of  (9  X  50+  no -[-400)^;^  =  12,000  lbs.  for  the  dead  load.  From  the  table  we  find  the  live 
load  end  reaction  to  be  66,500  lbs.,  making  a  grand  total  of  78,500  lbs.  At  250  lbs.  per  square 
inch  this  requfres  314  square  inches  of  bearing,  requiring  a  bed-plate  18  inches  square.  These 
bed-plates  when  placed  on  the  masonry  within  6  inches  of  the  face  of  the  masonry  under  the 
coping  will  be  50  feet  centre  to  centre,  and  the  over-all  length  of  the  girder  will  be  51  feet 
6  inches.  Our  assumption  of  50  feet  centre  to  centre  of  end  bearings  is  therefore  correct. 
The  weight  of  iron  will  be  changed  slightly,  as  /  in  the  formula  is  the  length  over  all  and  we 
have  used  it  as  the  centre  to  centre  length  for  convenience  in  the  preliminary  calculation  of 
the  end  shear.    The  correct  end  shear  will  be  as  follows : 


281.  The  Economical  Depth  of  a  plate  girder  may  be  closely  approximated  by  the 

following  formulae ; 

1st.  If  the  moment  of  resistance  of  the  web  is  neglected. 


Let  m  =  centre  moment  in  inch-pounds  from  dead  and  live  loads ; 
X  =  depth  of  girder  in  inches  ; 
J/  =  weight  of  girder  in  pounds ; 

/=  allowable  fibre  stress  =  7200  lbs.  per  square  inch  on  gross  area; 
/  =  thickness  of  web  =  f  inch ; 
/  =  length  of  girder  in  feet. 

Then,  as  the  weight  of  the  flanges  when  the  flange  plates  are  cut  off  at  the  theoretical  points 
may  be  assumed  as  eight  tenths  of  their  weight  in  case  they  were  the  same  area  end  to  end  of 


From  dead  load  (9  X  51-5  +  no  -f  400)-''^  =  12,200  lbs. 
"     live     "  ;=  66,500  " 


girder,  we  have 


y 


and 


dy 


10 


It  —  1.6 


nt    10.  I 


dx 


3 


-7- .--/.-,  =  o, 
/    3  ^ 


*  Ine  term  i.o— : .  — ,  /  vroula  be  a .  — 1.  — /  if  the  flatiKes  were  of  constant  area  end  to  end. 

Xf  Xf  5  .  - 


MODERN  FRAMED  STRUCTURES. 
whence 


/  in 


which  would  give  for  the  girder  assumed  79  inches  as  the  economical  depth. 
2d.  If  tlic,  moment  of  resistance  of  the  iveb  is  not  neglected. 

Using  the  same  notation  as  before,  and  also  assuming  that  the  net  moment  of  resistance 
of  the  web  results  in  an  allowable  reduction  of  each  flange  area  by  an  amount  equal  to  ant 
eighth  of  the  gross  area  of  the  web,  we  have 

j^/^.-./+i.6^.--./-g./^.- 

6      10  VI    10  , 

y  —  -tx  .  —  .  /  +  1.6^  .  —  .  /, 


and 


or 


and 


dy  3^  10,  .  m  10, 
dx      4     Z  fx  I 


\  fx^ 


m 

7r 


which  would  give  92  inches  as  the  economical  depth. 

The  quantities  which  may  vary  with  the  depth  and  are  neglected  in  the  above  form^^de 
are  the  weights  of  stiffeners,  sway  bracing  between  girders,  and  the  web  sjDlice  plates.  These 
would  have  little  effect  if  the  stiffeners  were  of  a  constant  size  and  placed  a  constant  propor- 
tion of  the  depth  apart.  Whatever  effect  these  neglected  quantities  would  have  would  tend 
to  reduce  the  depth. 

282.  Working  Rule  for  Economic  Depth. — A  very  convenient  rule  for  determining  approximately  the 
economical  depth  of  girder  in  the  case  when  the  moment  of  resistance  of  the  web  is  neglected  is  to 
select  the  depth  at  which  the  area  of  the  web  plate  is  equal  to  the  combined  area  of  the  two  flanges  if  the 
flanges  are  of  constant  area  end  to  end,  or  eight  tenths  of  the  combined  area  of  the  two  flanges  at  the  centre 
of  the  girder  if  flange  plates  are  used  and  stopped  of?  at  the  theoretical  lengths.  If  the  web  is  taken  mto 
account  at  its  full  value  in  resisting  the  bending  moment,  select  the  depth  at  which  three  quarters  of  the  area 
of  the  web  plate  is  equal  to  the  combined  area  of  the  two  flanges,  if  the  flanges  are  of  constant  area  end  to 
end  of  girder,  or  eight  tenths  of  the  combined  area  of  the  two  flanges  if  flange  plates  are  used  and  stopped 
off  at  the  theoretical  lengths.  In  the  latter  case  the  area  of  the  flanges  includes  the  web  equivalent,  or  is 
equal  to  the  bending  moment  divid^!^b|pthe  product  of  the  depth  and  unit  stress. 

283.  Determination  of  the  Flange  Areas. — For  the  span  under  consideration  the  dead 
load  centre  moinent  for  one  girder  is         ^  ^'^  —  152,200  ft.-lbs.,  and  the  live  load  centre 


The  term  i  6—  .  —  .  /  would  be  2 .  —  .  — /if  the  flanges  were  of  constant  area  end  to  end. 
-«■/     3  3 


THE  PLATE  GIRDER. 


301 


moment  taken  from  the  table  is  744,400  ft. -lbs.  As  the  usual  practice  is  to  neglect  the  effect 
of  the  web  in  resisting  the  bending  moment,  and  as  shown  previously  in  this  chapter  it  being 
very  difficult  to  make  a  web  splice  which  will  effectively  resist  bending,  the  influence  of  the 
web  will  be  neglected  ni  the  design  of  this  girder.  The  formulae  for  economical  depth 
give  79  inches,  but  in  order  to  make  allowance  for  the  possible  errors  in  the  assumptions  6  feet 
will  be  taken  for  the  depth.  Assuming  the  depth  centre  to  centre  of  gravity  of  the  flanges  as 
71  inches,  we  get  25,700  lbs.  as  the  dead  load  flange  stress  and  125,800  lbs.  as  the  live  load 
flange  stress.    The  required  net  area  of  the  bottom  flange  is  then  found  as  follows : 

18.5  square  inches. 


Two  6"  X  4"  X  xV  angles  =  9.5  square  inches  net  area; 
"     14"  X  f"  plates  =  9.0  square  inches  net  area. 

The  centre  of  gravity  of  this  flange  section  is  .33  inch  from  the  backs  of  the  angles,  making  the  depth 
centre  to  centre  of  gravity  of  the  flanges  71.34  inches.  This  is  near  enough  to  the  assumed  depth  to  require 
no  change  in  the  flange  area. 

The  rivets  are  seven  eighths  of  an  inch  in  diameter,  and  the  net  area  is  found  by  deducting 
one  hole  one  inch  in  diame'ter  from  each  angle  and  two  holes  from  each  plate,  it  being  assumed 
that  the  rivets  are  staggered  in  the  angles. 

284.  The  Lengths  of  Flange  Plates  are  usually  determined  as  follows : 
The  curve  of  maximum  bending  moments  is  assumed  to  be  a  parabola  as  shown  in  Fig. 
345,  an  assumption  which  is  in  error  about  three  per  cent  in  maximum  actual  effect,  and  the 
curve  plotted  with  a  middle  ordinate  representing  the  maximum  bending  moment. 


Fig.  345. 


The  corresponding  moments  of  resistance  of  the  plates  and  angles  ,  ,  and  .^3)  are 
then  laid  off  on  the  figure  to  the  same  scale,  and  the  lengths  of  the  plates  determined  by  the 
intersection  of  the  limiting  lines  A  and  B  with  the  curve.    Or  an  easy  way  of  determining 


125,800       7,500  =  16.8  [ 
The  flange  will  be  made  of 


MODERN  FRAMED  STRUCTURES. 


these  lengths,  assuming  the  curve  to  be  a  parabola  and  also  that  the  areas  of  all  the  members 
are  at  the  centre  of  gravity  of  the  flange  as  a  whole,  is  as  follows,  and  is  based  on  the  law  of 
the  parabola : 

Let  A  =  total  flange  area ; 

a^,  a^, . .  a„  =  areas  of  the  plates,  the  subscript  denoting  the  number  of  the  plate  from 
the  outside ; 
. . .  x„  =  length  of  the  plate ; 

/  =  length  of  girder  centre  to  centre  bearings. 


Then 


4-  ^r,  +  . . . 


^  Example. — In  the  above  girder  the  length  of  the  first  flange  plate  would  be 

I  -  50]/ ~~  =  24.66  ft., 

^  and  the  length  of  the  second  flange  plate  would  be 

^i^i!  504/-^  =  34.88  ft.. 


'is 


^  i  Whichever  of  the  above  methods  is  used,  it  is  customary  to  add  two  feet  to  the  theoretical 

V^"^  ^  length  of  the  plate  which  lies  next  to  the  angles,  and  one  foot  to  all  others,  in  order  to  get 

'  ^       "1r*  cr^jji  some  rivets  beyond  the  point  where  the  plate  is  necessary. 

,  285.  The  Thickness  of  the  Web  and  Size  of  Flange  Angles. — It  is  customary  in 

'     railroad  work  to  make  the  least  allowable  thickness  of  web  three  eighths  of  an  inch,  and  to 
^  limit  the  vertical  shearing  stress  on  the  vertical  section  of  the  web  plate  to  5000  lbs.  per 

square  inch.    Thus  in  the  girder  assumed  the  maximum  shear  is  78,700  lbs.,  and  the  area  of 
\  a  web  plate  three  eighths  of  an  inch  thick  is  27.0  square  inches,  giving  a  shearing  stress  of 

2900  lbs.  per  square  inch.    A  web  plate  three  eighths  of  an  inch  thick  will,  therefore,  satisfy 
^  this  requirement.    There,  is,  however,  another  question  involved,  and  that  is  that  there  must 

^-  be  enough  bearing  area  provided  for  the  rivets  through  the  flange  angles  and  web  plate  to 

^  I  ^  enable  them  to  transfer  the  flange  stress  to  the  flanges  without  exceeding  the  allowed  pressure 

«>i     \^    \.  per  square  inch  or  spacing  the  rivets  less  than  the  minimum  limit  of  three  diameters  centre  to 

^  '  ^  (.gpj-rg     In        girder  in  question  the  maximum  shear  is  78,700  lbs.,  and  from  eq.  (17)  the 


1^ 


j;  rivet  spacing  is 


N' 

X.  5 


^  "  \  ^      where  /  =  distance  centre  to  centre  of  rivets;  r  =  the  bearing  value  of  one  rivet  on  the 
;    web  plate,  in  this  case  3940  lbs.  {=  12,000  X  f  X  f ) ;     =  the  distance  from  the  rivet  lines 
^  I       ^  ^  bottom  flange  angles  to  the  rivet  lines  in  top  flange  angles,  which  in  the  girder  under 

^      ^  consideration  equals  68  inches ;  and  .S  =  the  shear.    Introducing  the  proper  values  into  the 


^  >^  formula. 


<   'K''^^^  *     3940  X  68 


I  ii  »  ■  ^  ^ 
11  " 


THE  PLATE  GIRDER.  303 

If  p  had  been  less  than  2|  inches,  or  three  times  |,  it  would  have  been  necessary  to  thicken  the  web 
plate,  or  use  flange  angles  in  which  two  rows  of  rivets  could  be  used  as  shown  in  Fig.  346. 

Thus,  suppose  the  shear  had  been,  for  the  case  under  consideration,  130,000  lbs.  This  would  still  have 
been  within  the  limits  of  the  allowed  shearing  stress  on  the  web  plate,  but  the  pitch  of  the  rivets  for  a  three- 
eighths  plate  would  be 

.     3940  X  68         ,  . 
p  =  —  =  2.06  in., 

1 30,000 


which  is  less  than  is  allowable.    To  find  the  thickness  of  web  required  for  flange  angles  in  which  oniy  ont 

I 

I 


) 

J 

4 

Fig.  346. 


T  h 

line  of  rivets  can  be  used,  we  must  solve  the  general  formula  p  =  — ^  for  r.   The  value  of  r  would  be 

ps 

Introducing  the  proper  values,  using  2f  as  the  value  of  p,  we  get 


X  I  "^o  000 

r  =  ~  —  =  5019  =  12,000  X  i  X  /,      and      /  =  .478  in.  or  J  in. 

60 

If  we  used  6"  x  6  '  angles  in  which  two  rows  of  rivets  could  be  put  through  each  of  the  flange  angles 
we  would  get,  bearing  in  mind  that  /i  for  this  case  is  65.5  in., 

7880  X  65.5 

p  =   —  =  3-97  >n- 

1 30,000 

The  use  of  the  larger  angles  would  save  some  material,  but  would  increase  the  cost  of  manufacture 
owing  to  the  increased  inimlier  f)f  rivets  to  be  driven. 


286.  The  Riveting  of  Plate  Girders. — It  is  necessary  in  designing  girders  to  provide 
sufificient  rivets  for  the  transfer  of  the  external  loads  or  forces  to  the  ^^/eh  plate,  and  to  so 
rivet  the  flanges  to  the  web  and  the  various  sections  of  the  web  to  each  other  that  the  rivets 
in  no  case  will  be  subjected  to  excessive  stresses.  When  the  flange  plates  and  angles  are  over 
50  ft.  long,  which  is  about  the  limiting  lengths  of  material  now  obtainable  at  standard  prices, 
it  also  becomes  necessary  to  splice  them  and  the  number  of  rivets  required  must  also  be 
determined  for  this  case. 

287.  The  Transfer  of  Concentrated  External  Loads  or  Forces  to  the  Web. — The 
usual  method  of  transferring  an  external  load  to  the  web  is  to  rivet  distributing  angles,  which 
fit  tight  against  the  flange  angles  at  the  point  of  application  of  the  load,  to  the  web,  using 
enough  rivets  to  resist  the  load  without  exceeding  the  limiting  values  of  shear  or  bearing.  In 


304 


MODERN  FRAMED  STRUCTURES. 


the  case  of  the  ordinary  wheel  load  of  a  locomotive  it  is  considered  to  be  distributed  over  a 
length  of  flange  of  about  3  ft.  by  the  rails,  and  no  provision  need  necessarily  be  made  for  such 
a  case  further  than  using  sufficient  rivets  through  the  flange  angles  to  transfer  it  directly  to 
the  web  without  distributing  angles.  In  the  usual  case  of  a  deck  plate  girder  the  only  con- 
centrated load  which  need  be  considered  worthy  of  distributing  angles  are  the  abutment 
reactions  for  which  distributing  angles  are  necessary.  The  usual  method  is  as  shown  in  Fig. 
346,  in  which  a  pair  of  angles  is  riveted  to  the  web  over  each  end  of  the  bed-plates.  These 
angles  should  have  a  tight  fit  against  the  horizontal  legs  of  the  lozuer  flange  angles.  The 
number  of  rivets  required  for  the  end  shear  in  the  girder  here  considered  would  be  78,700  -7- 
3940  =  20.  It  is  customary  to  divide  this  number  equally  between  the  two  pairs  of  angles, 
but,  owing  to  the  deflection  of  the  girder  bringing  the  greater  load  on  the  inside  pair, 
it  is  better  to  provide  more  than  this  in  the  inner  pair,  putting,  however,  at  least  one  half  of 
the  total  number  required  in  the  outside  pair  or  those  at  the  extreme  end  of  the  girder.  The 
size  of  these  angles  need  not  be  greater  than  is  required  to  insure  no  greater  bearing 
pressure  between  them  and  the  flange  angles  than  12,000  lbs.  per  square  inch.  The  usual 
practice  is  to  make  the  outstanding  leg  about  i  inch  less  than  the  horizontal  leg  of  the  flange 
angles  and  the  leg  against  the  web  3  or  3^  in.,  and  the  minimum  thickness  f  in. ;  the  bearing 
requirement  determining  the  thickness. 

288.  The  Transfer  of  the  Flange  Stresses  to  the  Flanges. — There  are  two  cases  to 
be  considered,  as  follows:  ist.  When  the  web  plate  is  assumed  to  have  no  moment  of  resist- 
ance and  to  transfer  shear  only ;  2d.  The  correct  assumption  where  the  web  is  considered  at 
its  actual  value  in  resisting  both  moments  and  shears. 

1st.  The  Transfer  of  the  Flange  Stresses  to  the  Flanges,  neglecting  the  Moment  of  Resist- 

rJi 

ance  of  the  Web  Plate. — From  Art.  279  we  get  the  formula  />  =  -^,when  /  =  pitch  or 

the  horizontal  distance  centre  to  centre  of  rivets  through  the  flange  ;  r,  the  value  of  the  rivet 
in  double  shear  or  bearing  value  on  the  web  plate  at  the  limiting  allowed  stresses ;  h,  the 
distance  from  the  line  of  rivets  through  the  web  in  the  top  flange  angles  to  the  line  of  rivets 
through  the  bottom  flange  angles ;  and  5,  the  vertical  shear  at  any  point  in  the  girder.  A 
brief  explanation  of  the  formula  will  be  given.  The  increment  of  flange  stress  or  the  amount 
of  flange  stress  which  must  be  transferred  to  the  flange  by  one  rivet  at  any  point  in  the  girder 
is  dependent  on  the  shear  at  that  point  and  the  horizontal  distance  centre  to  centre  of  rivets. 
The  shear  acting  with  a  lever  arm  ^  produces  an  increment  of  bending  moment  which  must 
be  resisted  by  a  moment  equal  to  it  and  equal  to  the  increment  of  flange  stress  acting  with  a 
lever  arm  of  h.  This  increment  of  flange  stress  is  transferred  from  the  web  to  the  flange  by 
the  rivet,  and  therefore  the  stress  on  the  rivet  must  be  equal  to  this  increment.  We  may 
then  write 

rk 

Sp=  rk,       or       p  = 


Now  as  the  total  shear  practically  increases  uniformly  from  the  centre  to  the  end  of  the 
girder,  we  can  easily  find  the  spacing  required  at  any  point.  By  finding  the  spacing  required 
at  the  end,  at  one  or  two  intermediate  points,  and  at  the  centre,  we  can  readily  sketch  in  a 
curve  as  shown  in  Fig.  347,  from  which  the  required  spacing  at  any  point  may  be  found 
graphically.    For  the  girder  we  have  under  consideration  the  spacing  required  at  the  end  is 

3940  X  68 

P  =   =  3-4  ins., 

^        78,700  ' 

the  spacing  required  at  the  quarter  point  is 


THE  PLATE  GIRDER. 


305 


_  3940  X  68 
^  ~  48,600 
and  the  spacing  required  at  the  centre  is 

3940  X  68 

P  ^ 


=  5.5  ins., 


=  14.4  ins. 


i  8,600 

Plotting  these  results  and  sketching  a  curve,  as  in  Fig.  347,  we  are  able  to  find  the  spac- 
ing required  at  intermediate  points.    The  usual  spacing  is  in  even  inches  or  half  inches,  and 


Fig.  347. 

the  maximum  allowable  space  is  6  ins.  for  practical  reasons.  From  the  diagram  it  is  easy  to 
determine  the  points  where  the  spacing  may  be  changed  without  exceeding  the  allowed  stress 
on  the  rivets. 

2d.  T/ie  Transfer  of  the  Flange  Stresses  to  the  Flanges  iising  the  Web  Plate  at  its  True 
Value. — The  difference  between  this  case  and  the  one  just  considered  is  that  in  the  former 
case  the  entire  bending  moment  is  assumed  to  be  resisted  by  stresses  in  the  flanges  alone,  and 
therefore  requiring  rivets  enough  to  transfer  this  stress  from  web  to  flange,  while  in  the 
present  case  only  such  an  amount  of  stress  is  transferred  as  actually  goes  to  the  flanges.  The 
amount  of  bending  moment  resisted  by  the  web  is  such  a  proportion  of  the  total  bending 
moment  at  any  given  section  as  the  moment  of  resistance  of  the  web  is  to  the  total  moment 
of  resistance  of  the  girder  at  that  section,  and  hence  the  amount  of  stress  to  be  transferred  to 
the  flanges  is  less  than  in  the  former  case  by  the  amount  of  flange-stress  equivalent  that  the 
web  takes.    It  has  been  shown  in  Art.  272  that 


M 


.'.  ^  =  equivalent  flange  stress  =  (•^"^~  ^)-/* 
Of  this  total  equivalent  flange  stress  the  part  7^ only  is  actually  transferred  to  the  flanges 


by  the  rivets;  hence  the  rivet  spaces  in  this  case  would  be  \  —  / /,  where  />  is  the  pitch, 

\  / 

or  distance  from  centre  to  centre,  of  rivets  determined  by  neglecting  the  efYcct  of  the  web 
plate.  When  there  is  a  series  of  flange  plates  stopped  off  at  the  theoretical  points,  a  section 
of  the  girder  near  the  end  of  a  plate  would  show  a  larger  flange  area  than  was  really  effective, 
and  the  corresponding  rivet  spacing  would  be  less  than  is  actually  required.    This  would  be 


3o6 


MODERN  FRAMED  STRUCTURES. 


an  error  on  the  side  of  safety.  The  correct  way  to  determine  the  rivets  for  tht<;  case 
would  be  to  assume  iox  F -\- —  the  theoretical  flange  area  required  at  the  section,  and  for  F 

o 

the  difference  between  this  theoretical  area  and  --. 

o 

289.  The  Combination  of  the  Vertical  Stress  on  the  rivets  from  the  external  loads 
and  the  horizontal  stress  from  the  bending  moment  will  now  be  considered.    The  maximum 
wheel  load  is  15,000  lbs.,  and  this  is  assumed  to  be  uniformly  distributed  by  the  rail  over  the 
space  occupied  by  three  cross-ties  which  is  about  42  inches.    Then  the  vertical  stress  on 
15,000 

each  rivet  would  be  .  p  =  3S7p,  approximately. 

42 

s  .p 

The  horizontal  stress  would  be  as  found  before,  r  =  -7—,  and  the  resultant  stress  from 

n 

these  two  would  be 


=  y^(357./r +  from  which         ^  ,27,400+ 


In  the  case  of  the  girder  under  consideration  —  3940  and  h  ~  68,  so  that  the  formula  re 
duces  to 


15,523,600 


127,400  + 


V68/ 


For  sections  taken  five  feet  apart  we  have 


s  = 

78,700, 

p  = 

3-25, 

spacing  at  end ; 

s  = 

66,700, 

p  = 

375» 

spacing   5  ft.  from  end  ; 

s  = 

54,600, 

p  = 

4-5, 

spacing  10  ft.  from  end  ; 

s  = 

42,700, 

/•  = 

5-5, 

spacing  15  ft.  from  end; 

S  — 

30,700, 

6.9, 

spacing  20  ft.  from  end  ; 

S  — 

18,600, 

/  = 

875. 

spacing  25  ft.  from  end. 

calculations,  the  term 


In  the  above  the  live  load  shear  is  assumed  to  increase  uniformly  from  centre  to  end, 
which  is  a  small  error  on  the  side  of  safety.  When  the  stiffener  angles  are  spaced  greater  dis- 
tances apart  than  three  feet,  the  rivet  spacing  should'  be  determined  by  this  method.  By  the 
correct  method  of  determining  rivet  spacing,  or  that  method  which  includes  the  web  in  the 

which  appears  in  the  denominator  would  be  |  ipj^^y^  j 

In  the  fifty-foot  girder  the  spacing  of  the  rivets  through  the  top  flange  angles  and  web 
plate  would  be  three  inches  until  six-inch  spaces  are  sufficient,  and  then  six  inches  the 
remainder  of  the  distance.  In  the  bottom  flange  the  same  rule  would  be  followed,  but  in  thiy 
case  it  will  be  noted  that  the  six-inch  spaces  will  begin  nearer  the  end  of  the  girder.  The 
point  when  six-inch  spaces  are  sufficient  in  the  lower  flange  would  be  determined  by  Fig.  347, 
and  where  they  may  begin  in  the  top  flange  by  Fig.  348. 


TH£  Plate  QiRD£:k. 


This  spacing  is  taken  because  it  will  enable  us  to  put  the  rivets  in  the  bottom  flange 
angles  on  the  same  vertical  lines  as  those  in  the  top  flange  angles,  and  thus  simplifies  the 
manufacture,  at  the  same  time  giving  us  a  larger  margin  of  safety  in  the  top  rivets,  and  also 
avoids  unnecessary  punching  and  weakening  of  the  lower  flanges.    The  number  of  rivets  will 


Fig.  348. 


be  about  the  same  as  if  we  had  used  the  spacing  indicated  by  Fig.  348  for  both  flanges,  which 
is  the  practical  alternative.  The  rivets  through  the  flange  plates  and  angles  are  always  stag- 
gered as  much  as  possible  with  those  through  the  web  and  flange  angles.  The  pitch  is  made 
six  inches  everywhere  except  near  the  ends  of  the  plates,  where  a  few  spaces  of  three  inches 
are  used. 

290.  Rivets  in  Web  Splices- — When  the  web  plate  is  not  calculated  as  resisting  any  of 
the  bending  moment,  the  web  need  be  spliced  for  vertical  shear  only.  Any  assumption  which 
may  be  made  does  not  relieve  the  web  of  its  true  stress  from  the  bending  moment,  so  that  it 
is  only  at  the  web  splices  that  the  flanges  are  subject  to  the  stress  that  this  assumption  indi- 
cates. At  this  point  the  excess  of  material  which  is  put  in  the  flanges  by  this  method  acts  as 
a  web  splice,  transferring  the  part  of  the  bending  moment  which  the  web  plate  does  resist 
across  the  spliced  section  of  the  web  plate.  The  ordinary  web  plate  splices  consist  of  a  pair 
of  plates  Y^g.  to  f  of  an  inch  thick  for  a  f-inch  web  with  a  single  vertical  line  of  rivets  on  each 
side  of  the  joint.  The  rivets  in  these  plates  are  figured  to  transfer  the  maximum  shear  at  this 
point  without  exceeding  the  allowed  shearing  stress  per  square  inch  on  rivets  or  the  allowed 
bearing  stress  per  square  inch  on  the  web  plate.  In  the  fifty-foot  girder  the  web  is  spliced 
about  8  feet  4  inches  from  the  centre,  where  the  shear  is  39,000  lbs.  The  bearing  value 
of  one  rivet  being  the  limiting  value  anti  equal  to  3940  lbs.,  ten  rivets  on  each  side  of  the 
splice  are  required.  In  order  not  to  exceed  six-inch  pitch,  eleven  rivets  on  each  side  would 
be  used. 

An  example  of  the  proper  method  of  splicing  webs  when  the  web  is  taken  into  considera- 
iion  in  determining  the  flange  areas  is  given  in  Art.  275. 

291.  Rivets  in  Flange  Splices. — It  is  customary,  and  also  economical,  when  the  flange 
section  must  be  spliced,  to  splice  only  one  piece  at  a  time.  For  example,  suppose  the  plates 
and  angles  in  the  flanges  of  the  fifty-foot  girder  were  longer  than  they  could  be  rolled,  the 
splice  would  be  made  as  shown  in  Fig.  349.  The  flange  plates  are  cut  on  the  lines  BB  and  CC 
respectively,  and  one  flange  angle  is  cut  at  AA  and  the  other  at  EE.  The  rivets  through  the 
splice  angles  and  plate  between  FF  and  AA,  AA  and  BB,  BB  and  CC,  CC  and  EE,  and  EE 
and  GG  must  in  each  case  equal  in  value  the  largest  piece  cut,  which  in  this  case  is  one  angle. 


3o8 


MODERN  FRAMED  STRUCTURES. 


The  value  of  this  angle  is  4.75  X  8200  —  39,000  lbs.,  which  requires  eleven  |  rivets  in  single 
shear.  The  value  of  one  f  rivet  is  3660  lbs.  (=  6000  X  .61).  The  sketch  shows  at  least 
twelve  rivets  in  each  case  in  single  shear.  The  net  area  of  the  splice  plate  or  of  the  two 
splice  angles  should  be  equal  to  the  net  area  of  the  largest  piece  cut.     Two  splicing 


F  A  B  C  EG 


— 


1 

A 

B 

C 

f 

1 

0 

0 

0 

0 

0 

0 

-t- 
1 
1 
1 

0 

0 

0 

-f=— == 

1  0 
1 

0 

1 

01 

0 

0 

0 

W- 

-Jr--_- 

 OC- 

0 

0 

0 

\o 

0 

0 

1 

1 
1 

0 

0 

1 

1  0 

0 

0 

0 

0 

0 

1 

_  1  

£"  A  B  C  E  G 

Fig.  349. 


angles  and  one  plate  are  generally  used.  Owing  to  the  three  eighths  limit  of  thickness 
generally  specified,  there  may  be  used  in  this  case  two  5  X  3i  X  |  angles  and  one  14  X  I 
plate. 

It  is  better  to  use  six-inch  pitch  for  rivets  in  the  tension  flange  splice  in  order  that  the 
staggering  of  the  rivets  may  be  as  effective  as  possible.  In  the  compression  flange  the  pitch 
may  be  made  as  small  as  three  diameters  of  the  rivet. 

292.  Stiffeners. — No  rational  method  has  yet  been  discovered  by  which  stiffeners,  or  the 
angles  which  are  riveted  to  the  web  to  prevent  its  buckling  under  stress,  can  be  proportioned. 
It  is  believed,  however,  that  if  they  are  spaced  at  distances  apart  a  little  less  than  the  depth 
of  the  web  plate,  and  made  to  consist  of  a  pair  of  angles  the  outstanding  legs  of  which  are 
one  thirtieth  of  the  depth  of  the  web  plate,  an  excessive  provision  has  been  made.  The 
stiffeners  under  this  rule  would  be  spaced  close  enough  to  resist  the  tendency  to  buckle  on  a 
line  making  forty-five  degrees  with  the  vertical,  and  the  combined  width  of  the  outstanding 
legs  would  make  a  column  fifteen  diameters  long.  This  is  below  the  limit  at  which  practical 
tests  have  shown  that  columns  begin  to  fail  by  flexure.  This  rule  would  also  agree  closely 
with  present  practice.  The  thickness  of  the  angles  should  not  be  less  than  five  sixteenths  of 
an  inch. 

Cooper's  specifications  require  that  stiffeners  must  be  used  at  distances  apart  about  equal  to  the  depth 
of  the  web  when  the  vertical  shearing  stress  per  square  inch  on  the  web  plate  exceeds  the  allowed  stress 

found  from  the  following  formula:  j  =    '"'"^^  ,  where  s  =  allowed  stress  and  //  is  the  ratio  of  the  depth 

I  -f  

3000 

of  the  web  plate  to  its  thickness,  or  ^,  where  /i  is  the  depth  of  the  web  plate  and  /  the  thickness  of  the 
web. 

Another  very  general  specification  is  that  stiffeners  must  be  used  whenever  the  vertical  shearing  stress 
per  square  inch  on  the  web  plate  exceeds  that  given  by  the  formula  s  —  — ,  where  s  =  the  allowed 

^  +  7» 

^ooor 

stress,  /=  depth  of  web  plate  or  distance  centre  to  centre  of  stiffeners,  and  /=  the  thickness  of  the  web. 
From  this  formula  the  distance  centre  to  centre  of  stiffeners  is  found.    In  the  standard  specifications  of  the 


THE  PLATE  GIRDER. 


Pennsylvania  Lines  another  good  method  is  specified.  It  is  that  stiffeners  must  be  used  whenever  the 
thickness  of  the  web  plate  is  less  than  one  fiftieth  of  the  clear  distance  between  the  vertical  legs  of  the  flange 
angles.  They  must  be  spaced  at  distances  apart  at  the  ends  of  the  span  not  greater  than  one  half  the  depth 
of  the  web  plate,  and  may  be  gradually  placed  further  apart  until  at  the  middle  of  the  span  they  are  not 
greater  than  the  depth  of  the  web  plate  centre  to  centre. 

It  is  not  customary  to  specify  the  sizes  to  be  used,  except  to  give  a  minimum  size  or  in  the  general 
clause  which  limits  the  sizes  of  material  to  be  used  in  any  part  of  the  structure. 

293.  Lateral  Bracing.- — The  lateral  bracing  of  a  plate  girder  bridge  is  now  usually  made 
of  angles  with  riveted  connections.  It  is  sufficient  to  use  only  one  system  of  bracing,  and  that 
in  the  plane  of  the  loaded  chord.  The  wind  pressure  on  the  bridge  is  taken  at  30  lbs.  per  square 
foot  of  the  bridge  as  seen  in  elevation,  and  this  is  combined  with  a  moving  load  of  300  lbs. 
per  linear  foot  of  track.  The  stresses  for  this  bracing  used  in  the  fifty-foot  span  are  given  on 
the  general  drawing  of  the  completed  design  shown  on  page  312,  and  are  calculated  by  the 
methods  explained  in  Part  I.    The  bracing  in  the  horizontal  plane  is  proportioned  for  com- 

/ 

pression  only  by  the  formula  13,500  —  60-,  which  allows  fifty  per  cent  greater  stresses  than 

8000 

that  allowed  by  the  usual  "  straight-line"  formula  substitute  for  Gordon's  formula,  -j^ — . 

^  8ooor' 

It  is  the  usual  practice  to  increase  the  allowed  stresses  in  wind  bracing  fifty  per  cent  over 
those  allowed  in  the  main  trusses. 

294.  The  Frames  at  the  ends  of  the  span  must  be  proportioned  to  resist  the  wind 
stresses,  assuming  all  the  forces  to  come  to  the  end  through  the  top  bracing.  Intermediate 
frames  should  be  used  at  distances  apart  not  over  sixteen  times  the  width  of  flange.  This 
rule  agrees  with  the  general  practice. 

295.  Bed-plates  should  always  be  made  thick  enough  to  insure  uniform  pressure  over 
the  masonry.  They  are  made  f  of  an  inch  thick  for  the  fifty-foot  span  under  consideration, 
v/hich  will  be  sufficient  as  the  pressure  of  the  girder  on  the  bed-plates  is  distributed  over  an 
area  of  18''  X  I2f",  leaving  less  than  3  inches  projection  unsupported. 

296.  There  are  Sole  Plates,  generally  of  the  same  thickness  as  the  bed-plates,  riveted 
to  the  bottom  flanges  of  the  girders,  and  at  the  expansion  ends  the  surfaces  of  contact  between 
bed  and  sole  plates  are  planed  to  insure  a  minimum  amount  of  friction. 

297.  Expansion  and  Contraction  in  a  plate  girder  bridge  of  a  span  less  than  75  feet  is 
U'^ually  provided  for  by  allowing  one  end  to  move  on  the  planed  surfaces  between  the  sole 
and  bed  plates  mentioned  in  the  previous  article,  the  other  end  being  fixed.  For  lengths 
over  75  feet,  rollers  not  less  than  two  inches  in  diameter  are  put  between  the  bed  and  sole 
plates  at  the  expansion  end.  When  any  bridge  rests  on  rollers,  it  is  important  that  the  press- 
ure should  be  distributed  as  uniformly  as  possible  over  the  rollers.  The  present  practice  of 
merely  putting  the  rollers  between  the  sole  and  bed  plates  without  any  attempt  being  made 
to  make  the  pressure  equal  on  the  rollers  is  not  correct,  as  has  been  explained  in  Chapter 
XVIII. 

298.  Anchor  Bolts,  usually  one  inch  in  diameter  for  all  plate-girder  spans,  are  put 
through  the  bed  and  sole  plates  and  run  about  six  inches  into  the  masonry.  They  are  either 
"  rag"  or  "  wedge"  bolts,  and  after  being  put  in  place  the  hole  in  the  stone-work  is  packed 
with  cement  or  sulphur.*  Slotted  holes  for  these  bolts  are  made  in  the  j<7/ir-plate  only,  at  the 
expansion  end,  to  allow  for  the  movement  of  the  girder  on  the  bed-plate.  There  are  usually 
two  of  these  bolts  through  each  bed  or  sole  plate,  or  eight  to  each  complete  single  track  span. 

299.  The  Width  centre  to  centre  of  the  girders  of  deck  plate-girder  spans  varies  from 
5  feet  to  10  or  12  feet  on  straight  track.  Cooper  specifies  a  minimum  width  of  6  feet  6  inches 
on  straight  track  and  that  the  girders  in  no  case  shall  be  less  than  3  feet  3  inches  from  the 


*  See  Fig.  292. 


3IO 


MODERN  FRAMED  STRUCTURES. 


centre  of  the  track.  A  good  rule  to  follow  is  to  space  the  girders  as  far  apart  as  the  depth 
of  the  girder  on  straight  track  and  increase  this  distance  by  the  middle  ordinate  of  the  curve 
on  the  bridge  for  curves,  with  a  minimum  distance  of  3  feet  3  inches  from  centre  of  track  to 
the  nearer  girder  for  all  cases. 

300.  The  Complete  Design  of  the  girder  which  has  been  referred  to  all  through  this 
discussion  is  given  on  page  312. 

301.  The  Design  of  Through  Plate  Girder  Spans. — If  the  cross-ties  are  supported 
on  the  inside  bottom  flange  angles  or  on  a  shelf  angle  as  shown  in  Fig.  268,  Chapter  XVI, 
the  girders  are  dimensioned  in  the  same  manner  as  for  a  deck  span.  If  shelf  angles  are  used, 
they  should  be  not  less  than  4  inches  by  3  inches,  and  f  thick,  with  the  4-inch  leg  horizontal. 
It  is  better  to  use  distributing  angles  under  the  shelf  angle  about  3  feet  apart,  to  transfer  the 
load  to  the  web,  than  to  depend  on  the  rivets  attaching  the  shelf  to  the  web,  owing  to  the 
necessarily  eccentric  loading  of  the  shelf  angles  producing  a  bending  moment  which  would 
bring  a  tensile  stress  on  the  rivets.  The  rivets  in  the  flanges  for  this  case  would  be  spaced  by 
the  diagram  Fig.  347,  as  there  are  no  external  loads  acting  on  the  flanges  to  be  transferred 
to  the  web.  If  the  cross-ties  are  supported  on  the  inside  bottom  flange  angles,  these  angles 
should  not  be  less  than  |  of  an  inch  thick  and  the  rivets  through  the  web  and  these  angles 
should  be  spaced  the  minimum  allov/able  pitch,  as  there  is  necessarily  a  tensile  stress  on  them 
from  the  eccentric  application  of  the  bearing  of  the  cross-ties  on  the  angles.  The  rivets 
through  the  top  flange  may  be  spaced  by  the  diagram  Fig.  347.  It  must  be  noted  that  there 
is  no  necessity  for  increasing  the  area  of  the  lower  flange,  except  when  necessary  in  order 
to  reach  the  minimum  allowed  limit  of  thickness  for  the  inside  angles  of  %  of  an  inch,  as  the 
stresses  are  increased  a  very  small  amount. 

If  an  iron  floor  system  of  floor-beams  and  stringers  is  used,  the  live  load  and  the  weight 
of  the  floor  and  floor  system  are  concentrated  at  the  points  of  attachment  of  the  floor-beams 
or  panel  points,  and  are  no  longer  distributed  over  the  length  of  the  girder  as  in  the  case  of 
the  deck  bridge.  The  panel  lengths  used  range  from  8  to  15  feet,  varying  with  the  length  of 
the  girder,  but  are  usually  about  10  feet.    In  Fig.  350  is  shown  a  longitudinal  sectional  view 


on  the  centre  line  of  a  through  plate-girder  span  with  an  iron  floor  system.  The  bending 
moment  and  shears  are  found  at  the  critical  points  A,  and  C,  and  the  flanges  dimensioned 
as  illustrated  for  the  deck  span.  The  bending  moment  varies  uniformly  from  A  to  B,  and 
from  B  to  C,  and  is  nearly  uniform  from  C  to  the  centre.  The  shear  is  constant  from  A  to  B^ 
B  to  C,  and  from  (7  to  the  centre.* 


*  This  is  not  strictly  true,  as  the  weight  of  the  girder,  which  is  a  small  proportion  of  the  load,  is  distributed  over 
the  entire  length  of  the  girder.  It  is  the  usual  practice  to  consider  the  weight  of  the  girder  concentrated  at  the  panel 
points  in  order  to  simplify  the  ^ork  of  calculation  and  as  the  error  is  insignificant. 


THE  PLATE  GIRDER. 


3" 


1 

B 

1 

The  rivet  spacing  is  easily  determined  after  the  critical  moments  and  shears  are  founds 
by  the  same  formula  as  was  used  for  the  deck  girder,  remembering  that  there  are  no  concen- 
trated external  loads  on  the  flanges. 

302.  Attachments  of  the  Floor-beams  and  Stringers. — There  must  be  enough  rivets 
in  the  connection  between  the  floor-beam  and  the  girder  to  transfer  the  end  shear  of  the 
floor-beam  to  the  girder.  The  rivets  are  in  single  shear.  If  they  are  "  field"  rivets,  it  is  cus- 
tomary to  put  25  to  50  per  cent  more  rivets  than  would  be 
used  if  the  rivets  were  to  be  shop-driven.  The  floor-beam 
is  often  attached  as  shown  in  Fig.  351,  in  which  case  there 
must  be  the  same  number  of  rivets  in  the  splice  plate  B 
to  splice  the  web  plate  to  the  gusset  plate  as  though  it 
were  simply  a  web  splice.  The  number  of  rivets  connect- 
ing the  gusset  plate  to  the  stiffener  angle  and  the  stiffener 
angle  to  the  web  must  be  sufificient  to  transfer  the  total 
end  shear  of  the  floor-beam.  In  the  attachment  of  the 
stringers  to  the  floor-beam  as  shown  in  Fig.  350  there  must 
be  enough  rivets  to  transfer  the  total  end  shear  of  one 

stringer,  counting  the  rivets  in  single  shear ;  and  if  there 

are  stringers  attached  on  each  side  of  the  beam  at  the  same  point,  there  must  be  enough  rivets 
to  transfer  the  maximum  combined  end  shear,  or  total  concentration  on  the  beam  at  that 
point,  to  the  floor-beam,  counting  the  rivets  in  double  shear  or  at  the  allowed  bearing  value  on 
the  web  of  the  beam. 

There  is  usually  no  end  floor-beam  in  through  bridges  of  this  kind,  the  stringers  resting 
on  the  masonry  direct  and  held  in  line  by  an  end  strut  which  runs  from  girder  to  girder  and 
which  also  attaches  to  the  end  gusset  plates. 

The  floor-beams  for  through  girders  of  this  kind  are  calculated  for  a  length  centre  to 
centre  of  end  supports  equal  to  the  width  centre  to  centre  of  main  girders.  The  stringers  are 
calculated  for  the  length  centre  to  centre  of  floor-beams,  or  in  case  of  the  end  stringers  where 
no  end  floor-beam  is  used  the  length  assumed  in  the  calculation  would  be  from  the  centre  of 
the  end  bearing  plate  to  the  centre  of  the  first  beam.  For  panel  lengths  less  than  15  feet 
the  stringers  are  usually  rolled  I  beams  and  are  always  proportioned  by  their  moment  of 
resistance. 

303.  The  Bracing  of  a  Through  Plate-girder  Span  differs  slightly  from  that  of  a 
deck  span,  inasmuch  as  in  this  case  provision  has  to  be  made  for  supporting  the  top  or  com- 
pression flange  against  side  deflection.  This  is  usually  done  by  means  of  gusset  plates  from 
the  floor-beam,  or  cross-strut,  in  the  case  where  no  iron  floor  system  is  used,  similar  to  the 
method  shown  in  Fig.  351.  Often  there  is  only  an  angle  brace  riveted  to  the  vertical  leg  of 
the  top  flange  angle  and  bent  out  so  as  to  attach  to  the  floor-beam  or  cross-strut  two  or  three 
feet  from  the  girder.  These  gussets  or  braces  and  also  the  floor-beams  or  cross-struts  should 
be  spaced  not  over  15  feet  apart.  The  lateral  bracing  proper  may  consist  either  of  adjustable 
rod  or  of  angle-iron  bracing  attached  to  the  main  girders.  It  is  customary  to  use  angle  bracing 
with  riveted  connections,  and  owing  to  their  being  of  greater  length  than  for  a  deck  span  they 
are  usually  made  to  cross  the  panel  on  both  diagonals,  each  piece  being  proportioned  to  take 
the  total  stress  in  tension. 

304.  The  Width  Centre  to  Centre  of  Girders  for  a  through  plate-girder  span  is  never 
less  than  12  feet  on  straight  track.  The  girders  should  be  spaced  so  as  to  give  at  least  7  feet 
clear  distance  from  the  centre  of  the  track  to  the  inside  edge  of  the  flange  plates  if  they 
extend  more  than  one  foot  above  the  rails,  in  order  to  give  ample  clearance  for  the  passage  of 
trains.  This  is  the  usual  requirement  now,  although  some  of  the  Western  roads  require  7 
feet  6  inches  from  the  inside  of  the  flange  plates  to  the  centre  of  the  track. 


ROOF  TRUSSES. 


313 


CHAPTER  XX. 
ROOF  TRUSSES. 

305.  The  Type  of  Truss  most  commonly  used  for  shop,  warehouse,  and  small  train-shed 
iron  roofs  is  what  is  known  as  the  Fink  truss,  shown  in  Fig.  353.    It  has  proved  to  be  a  very 


Fig.  353. 


satisfactory  and  economical  type  for  the  ordinary  lengths,  which  are  under  100  teet.  There 
are  many  conditions  which  may,  however,  affect  the  design  of  a  roof  ;  and  as  there  are  very 
few  reasons  why  any  special  type  of  truss  should  be  adopted,  it  may  be  generally  accepted 
that  the  best  truss  to  select  is  the  one  which  fulfils  the  special  conditions  the  best  and  at  the 
same  time  is  economical  in  material.  It  would  be  very  creditable  designing  to  select  a  truss 
in  which  the  lines  present  the  most  pleasing  appearance  if  the  fitness  of  the  truss  for  the 
purposes  for  which  it  is  to  be  used  is  not  impaired  thereby. 

For  the  longer  spans  the  sickle  roof  truss  and  the  three-hinged  arch  are  generally  used 
wherever  the  conditions  permit.  Either  of  these  designs  will  make  creditable  trusses.  For 
train  sheds  the  three-hinged  arch  is  preferred  before  all  others.  The  recently  constructed 
train  sheds  of  the  Pennsylvania  and  the  Philadelphia  and  Reading  railroads  at  Philadelphia 
and  Jersey  City  are  very  fine  examples  of  the  three-hinged  arch  construction.    (See  Fig.  65.) 

For  shop  construction  one  of  the  usual  requirements  is  a  horizontal  lower  chord  for  the 
purpose  of  supporting  shafting  or  trolley  runways,  and  in  these  cases  the  Fink  truss  is  usually 
employed  ;  but  any  other  form  with  a  horizontal  lower  chord  will  suffice. 

If  the  trusses  are  supported  on  iron  columns  instead  of  walls,  the  wind  force  is  transferred 
to  the  foundations  through  the  columns.  This  produces  a  bending  moment  in  the  columns 
which  is  a  maximum  at  the  top  and  must  be  transferred  to  the  truss,  a  requirement  which 
often  modifies  the  form  of  the  latter  if  economy  is  an  object.*  The  stresses  in  the  truss  are 
less  if  the  truss  is  deep  at  the  ends  ;  and  if  the  Fink  truss  is  used,  knee  braces  from  the  column 
to  the  first  joint  of  the  lower  chord  are  necessary.  If  an  ordinary  triangular  truss  is  used 
with  some  depth  at  the  ends,  knee  braces  are  not  used. 

The  slope  of  the  top  chord  is  usually  determined  by  the  kind  of  roof  covering  used.  This 
slope  must  be  steep  enough  to  allow  the  covering  used  to  be  of  the  best  service;  thus,  slate 
should  not  be  used  on  a  roof  having  a  slope  less  than  one  to  three  and  preferably  one  to  two. 
Tin  or  gravel  may  be  used  on  a  slope  of  one  to  twelve.  The  greater  the  pitch  the  greater  is 
the  area  of  the  roof  covering  required. 

306.  Riveted  or  Pin  Connected  Roof  Trusses. — Roof  trusses  are  usually  made  with 
riveted  connections,  this  being  the  cheaper  kind  of  construction  for  the  usual  short  spans  and 
small  truss  members.  There  are  cases,  however,  when  the  pin  connection  may  be  the  cheaper 
*  For  a  full  discussion  of  these  stresses,  see  Chap.  XXIX,  on  Iron  and  Steel  Mill  Building  Construction. 


3»4 


MODERN  FRAMED  STRUCTURES. 


or  more  advisable  construction.  When  the  span  is  long  or  very  heavy,  requiring  compara- 
tively large  members  in  the  trusses,  or  when  the  material  must  be  transported  a  great  distance 
from  the  shops,  there  may  be  a  saving  in  manufacture  or  in  transportation.  The  pin  connection 
may  also  be  used  to  avoid  any  field  riveting.  In  the  case  of  a  roof  over  a  building  in  which 
there  are  gases  which  have  a  very  marked  corrosive  effect  on  iron,  it  is  generally  advisable  to 
use  pin  connections,  because  the  members  of  this  kind  of  a  truss  expose  a  less  surface  to  cor- 
rosion than  do  the  thin  angle  irons  which  would  be  used  in  a  riveted  truss. 

Sometimes,  in  order  to  provide  for  corrosion  and  at  the  same  time  get  the  benefit  of  the  cheaper  con- 
struction of  the  riveted  truss,  a  minimum  thickness  is  specified  which  must  be  added  to  the  thicknesses  of 
all  material  as  required  by  the  stresses.  Thus,  if  of  'w\c\\  was  to  be  added  on  all  sides,  making  a  total 
addition  in  thickness  of  -jV  of  ^"  inch,  an  angle  iron  3"  x  2"  x  ^'  would  be  increased  of  a  square  inch 
or  25  per  cent  in  area,  while  a  round  rod  \\  inches  in  diameter  would  be  increased  only  ^  of  a  square  inch 
or  10  per  cent  in  area. 

307.  Ordinary  Roof  Coverings. — For  buildings  with  iron  roof  trusses  the  coverings  most 
commonly  used  are  corrugated  iron,  tin,  and  slate.  Corrugated  iron  and  slate  are  used  when 
the  slope  of  the  roof  is  greater  than  three  horizontal  to  one  vertical.  For  flatter  roofs  than 
this  they  are  liable  to  leak,  as  the  usual  joints  are  not  tight.  Tin  may  be  used  on  any  slope 
or  on  a  flat  roof.  Corrugated  iron  is  usually  laid  directly  on  the  purlins,  to  which  it  is 
attached  by  means  of  clips.  Owing  to  its  being  stiff  enough  to  span  a  distance  of  about  six 
feet,  sheathing  boards  are  not  necessary.  Tin  and  slate  are  usually  laid  on  sheathing,  a  layer 
of  roofing  felt  being  put  between  the  tin  or  slate  and  the  sheathing.  A  very  expensive  con- 
struction is  that  in  which  the  slate  is  supported  directly  on  iron  purlins,  which  must  generally 
be  about  lo^  inches  apart.    The  expensiveness  of  this  plan  is  due  to  the  weight  of  the  purlins. 

The  weight  of  any  kind  of  roof  covering  may  be  obtained  from  various  handbooks.  The 
sheathing  is  usually  assumed  to  weigh  four  pounds  per  foot,  board  measure.  The  thickness 
of  the  sheathing  is  determined  by  the  distance  apart  of  the  purlins,  between  which  it  must  be 
able  to  carry  the  weight  of  the  roof  covering  and  the  wind  or  snow  load,  usually  allowing  a 
fibre  stress  of  1500  lbs.  per  square  inch.    The  usual  thicknesses  are  f,  i^,  and  2  inches. 

308.  The  Loads  for  which  roofs  are  usually  proportioned  are  the  weight  of  the  roof 
covering,  the  sheathing  if  any,  the  iron  weight  in  trusses  and  purlins  and  the  snow  and  wind 
loads.  The  weight  of  the  covering  which  it  is  proposed  to  use  may  be  obtained  from  any 
good  handbook.  The  weight  of  the  sheathing  is  dependent  upon  its  thickness,  and  that  may 
be  determined  from  the  distance  centre  to  centre  of  purlins.    The  weight  of  the  iron  may  be 

/ 

closely  obtained  from  the  formula  tv  —  —  -|-  4.0,  where  w  —  weight  per  square  foot  of  covered 

horizontal  area  and  /  =  length  of  span.  This  weight  of  iron  is  only  for  the  case  of  the  roof 
which  carries  no  local  concentrated  loads.  The  wind  load  is  usually  assumed  at  30  lbs.  per 
square  foot  of  roof  surface.  The  snow  load  is  usually  taken  at  20  lbs.  per  square  foot  of 
horizontal  surface. 

In  the  calculation  of  trusses  with  curved  chords  it  is  the  usual  practice  to  find  the  stresses 
for  all  the  different  loadings  separately.  Thus,  the  stresses  would  be  calculated  for  the  wind 
on  the  side  of  the  truss  nearer  the  expansion  end,  and  for  the  wind  on  the  side  of  the  truss 
nearer  the  fixed  end  of  the  truss.  The  snow  may  be  figured  as  covering  the  entire  roof  or 
only  one  half,  and  even  in  special  cases  only  a  small  area  on  one  side.  It  is  not  generally 
assumed  that  the  maximum  wind  pressure  and  the  snow  load  can  act  on  the  same  half  of  the 
truss  at  the  same  time.  In  the  Fink  truss  a  partial  load  like  any  of  the  above  never  causes 
any  maximum  stresses,  so  that  it  is  customary  to  calculate  these  trusses  for  a  uniform  load 
over  the  entire  truss,  the  wind  and  snow  loads  combined  being  usually  assumed  at  30  lbs.  per 
square  foot  of  covered  area.  This  simplifies  the  calculations  very  much,  and  the  results  are  as 
nearly  correct  as  we  can  hope  to  arrive  at  by  any  assumed  exact  loading. 


HOOF  TRUSSES.  Z^S 

309.  The  Allowed  Stresses  per  square  inch  which  have  been  very  gcerally  adopted 
correspond  to  what  is  termed  a  factor  of  safety  of  four.  The  usual  allowed  stresses  per 
square  inch  on  iron  and  steel  in  roof  construction  are  given  in  the  following  table : 

Wrought-iron. 

Tension  shape-iron   12,000  lbs. 

"     rods  and  eyebars   15,000  " 

Maximum  fibre  stress  on  I  beams   12,000 

10,000 
 -p  flat  ends, 

1  + 


36,ooor' 

Compression  j  ^ 

 —  pin  ends. 

I  +  S  T 

1 8,ooor 

Shear  on  rivets  and  pins   75oo  lbs. 

Bearing  of  rivets  and  pins   15,000  " 

Bending  fibre  stress  on  pins   18,000  " 

Steel. 

Tension  (shapes)   15,000  " 

"       rods  and  eyebars   18,000  " 

Maximum  fibre  stress  on  I  beams   16,000  " 

Compression  20  per  cent  more  than  that  allowed  on  wrought-iron. 

Shear  on  rivets  and  pins   9000  lbs. 

Bearing  on  rivets  and  pins   l8,000  " 

Bending  fibre  stress  on  pins   22,500  " 

In  case  the  roof  was  subjected  to  a  load  from  cranes,  shafting,  etc.,  the  above  allowed 
stresses  would  be  reduced.  In  the  riveted  trusses  there  are  secondary  stresses  arising  from 
the  eccentric  application  of  the  stresses  on  the  various  members  which  are  not  provided  for 
further  than  to  reduce  them  to  the  minimum  possible  amount.  This  is  a  point  in  the  design 
of  riveted  trusses  which  should  always  have  careful  consideration. 

310.  The  Economical  Distance  Centre  to  Centre  of  Roof  Trusses  is  dependent 
principally  upon  the  relative  unit  stresses  in  the  purlins  and  the  main  trusses.  The  weight  of 
the  purlins  per  square  foot  of  covered  area  varies  almost  proportionally  to  the  distance  centre 
to  centre  of  trusses.  The  trusses  for  a  constant  length  of  span  would,  if  it  were  possible  to 
realize  the  small  areas  required  for  the  lighter  trusses,  weigh  the  same  per  square  foot  of  covered 
area  for  all  distances  centre  to  centre  of  trusses,  as  the  weight  of  the  trusses  would  in  this  case 
vary  as  the  load  on  them,  which  is  a  fixed  number  of  pounds  per  square  foot  of  covered  area. 
The  total  weight  per  square  foot  of  covered  area  for  the  purlins  and  trusses  would,  if  the 
above  were  true,  be  a  minimum  where  the  distance  centre  to  centre  of  trusses  is  a  minimum 
or  zero.  The  trusses  for  a  given  span,  however,  do  not  vary  directly  as  the  load,  but  they  are 
found  to  vary  in  weight  approximately  as  follows : 

,b 

w  =  a  -] — , 

X 


where  w  =  weight  of  trusses  per  square  foot  of  horizontal  surface  covered,  a  and  ^  are 
constants,  and  x  =  distance  centre  to  centre  of  trusses.     That  is,  a  part  of  the  weight  of 


3i6 


MODERN  FRAMED  STRUCTURES. 


trusses  per  square  foot  of  covered  area  remains  constant  for  any  distance  centre  to  centre  ol 
trusses,  and  the  rest  varies  inversely  as  the  distance  centre  to  centre  of  trusses  or  directly  as 
the  number  of  trusses.  The  constants  a  and  b  may  be  found  for  any  length  of  span  by  assum- 
ing various  widths  centre  to  centre  of  trusses  and  plotting  the  results,  using  zv  and  x  as  the 
variables.  Or  it  may  be  generally  assumed  that  the  width  centre  to  centre  of  trusses  for 
maximum  economy  in  material  is  about  one  fifth  of  the  span.  For  economy  in  manufacture  a 
greater  distance  between  trusses  is  necessary,  as  the  cost  of  manufacture  varies  almost  directly 
as  the  number  of  trusses. 

311.  The  Detail  Design  of  the  Purlins  and  Trusses  for  a  Roof. — Assume  the  trusses 
to  be  60  feet  centre  to  centre  of  end  bearings  and  spaced  20  feet  centre  to  centre.  The 
allowed  stresses  to  be  as  given  in  Art.  309.  The  covering  to  be  slate  on  two-inch  sheathing. 
The  truss  will  be  the  ordinary  Fink  truss  shown  in  Fig.  353,  with  a  depth  at  the  centre  of 
15  feet. 


Fig.  353. 

312.  Calculation  of  the  Purlins. — The  purlins  or  stringers  rest  on  the  trusses  at  the  joints 
of  the  upper  chord  and  serve  to  support  the  loads  on  the  roof  between  the  trusses.  As  the 
trusses  in  our  case  are  60  feet  centre  to  centre  of  end  bearings,  and  as  there  is  a  purlin  at  each 

joint  of  the  upper  chord  of  the  truss,  the  distance  between  purlins  is^('V^  15"  +  30")=  8.4  feet, 
nearly.  The  slate  covering,  felt,  and  sheathing  will  weigh  13  lbs.  per  square  foot.  The  purlin 
itself  we  will  assume  to  weigh  2  lbs.  per  square  foot,  and  the  wind  or  snow  load  is  taken  at  30 
lbs.  per  square  foot,  making  a  total  load  of  (13-1-2  +  30)8.4  =  378  lbs.  per  linear  foot  of 
purlin.    The  maximum  moment  on  the  purlin,  assuming  an  I  beam,  will  be 

— a 
20 

378  X  -5-  =  18,900  foot-lbs.,       or       226,800  inch-lbs. 

o 

From  the  formula  t^/  =~  vjc  get  ^=  -  .    From  any  of  the  rolling-mill  handbooks  we 

y,       ^     f     y,       .  ^  ^ 

can  get  the  value  —  directly  from  the  tables  giving  the  properties  of  I-beam  sections.  This 

term  —  is  called  the  Moment  of  Resistance  of  the  section.    Dividing  the  moment  226,800  by 

16,000,  the  allowed  stress  in  extreme  fibres  for  steel  I  beams,  we  get  14.2  for  the  moment  of 
resistance  of  the  required  steel  I  beam.  Using  the  sections  and  handbook  of  the  Carnegie 
Steel  Co.,  we  find  that  the  lightest  beam  which  has  the  required  moment  of  resistance  to  be 
an  eight-inch  beam  weighing  18  lbs.  per  foot. 

We  could  have  used  a  seven-inch  beam  weighing  20  lbs.  per  foot.  If  the  required  moment  of  resistance 
had  been  16.0  instead  of  14.2,  we  would  have  selected  the  beam  in  the  following  manner,  as  the  section  of  the 
beam  given  in  the  handbook  was  lighter  than  required.  Suppose,  as  is  the  case,  that  the  moment  of  resist- 
ance (R)  of  the  eight-inch  beam  given  in  the  table  was  14.4,  then  the  increase  in  R  necessary  would  be  1.6. 
As  the  increase  in  the  section  of  an  I  beam  is  accomplished  by  drawing  the  rolls  apart,  and  therefore  the 
addition  to  the  section  is  rectangular  with  one  side  equal  to  the  depth  of  the  beam,  in  this  case  eight  inches, 
and  the  other  side  equals  the  distance  which  the  rolls  were  drawn  apart,  and  the  increase  in  the  moment  of 
resistance  is  | .  M',  where  h  =  depth  of  beam  and  b  the  distance  which  the  rolls  were  separated.    This  in  the 


ROOF  TRUSSES. 


case  of  the  eight-inch  beam  cited  will  be  \  .  bh  .  h  =  \  x  increase  in  area  of  beam  x  8  =  1.6,  or  the  increase 
in  area  of  the  eight-inch  beam  equals  1.2  square  inches  for  an  increase  in  the  moment  of  resistance  of  1.6, 
which  is  equivalent  to  an  increase  in  weight  of  the  beam  of  nearly  4  lbs.  per  foot.  Hence  an  eight-inch 
beam  weigiiing  22  lbs.  per  linear  foot  would  have  been  required. 

The  purlins  are  often  made  king  or  queen  post  trusses,  but  beams  for  ordinary  lengths  will  be  lighter 
and  cheaper. 

313.  The  Detail  Design  of  the  Sixty-foot  Roof  Truss  can  now  be  made.  The 
dead  weight  of  the  covering,  sheathing,  purlins,  and  trusses  will  oe  assumed  as  follows : 

Slate  and  felt  5    lbs.  per  square  foot ; 

Sheathing  (2-inch)  8      "     "       "  " 

Iron  (=         +  4)  6.4  "  " 

making  a  total  dead  weight  of  19.4  lbs.  per  square  foot.    For  convenience  it  will  be  assumed 

20  lbs.  per  square  foot  of  covered  area.    The  snow  and  wind  loads  will  be  assumed  as  usual  to 

be  30  lbs.  per  square  foot  of  horizontal  area  covered.    The  load  at  each  truss  joint  of  the  top 

chord  (assuming  the  iron  weight  of  the  truss  to  be  concentrated  there  also)  will  be,  combining 

all  the  loads,  50  X  7^  X  20  =  7500  lbs.,  where  50  equals  the  total  load  per  square  foot  of 

covered  area,  7^  the  horizontal  distance  in  feet  between  purlins,  and  20  the  distance  in  feet 

centre  to  centre  of  trusses.    The  stresses  in  the  truss  may  then  be  found  by  any  of  the 

methods  given  in  Part  I.    Owing  to  the  similarity  of  trusses  of  this  kind  it  is  more  convenient 

to  work  out  the  stresses  in  each  member  for  a  panel  load  of  one  pound  and  tabulate  the 

results  as  shown  in  the  table  of  coefficients  on  the  following  page,  from  which  the  stresses 

may  be  derived  quickly  by  the  use  of  the  slide-rule.    The  total  stresses,  allowed  stresses  per 

square  inch,  the  length  centre  to  centre  of  support  of  compression  members,  the  least  radius 

of  gyration  of  compression  members,  the  required  area  in  square  inches,  and  the  make-up  and 

area  of  the  various  members  of  the  truss  are  given  in  the  following  table. 

p 


Radius 

Unit  Stress 

Area 

Member. 

Total  Stress. 

Length. 

of 

Gyration. 

Allowed. 

Required. 

Make-up  of  Members. 

Area  Used, 

I 

+  58,700 

lOl" 

1-25 

8,500 

6.91 

Two  5"  X  3"  Zs,  34.5  lbs.  per  yard 
"     5"  X  3"  Zs,  34-5  " 
"     5"  X  3"  Zs,  34-5  " 
"     5"  X  3"  ZS,  34.5  " 
"     2"  X  2*"  X  r  Zs 
"     2"  X  2i"  X  i"  Zs 
"     2"  X  2i"  X  i"  Zs 
"    2"  X  3"  X  i"  Zs 
"    2"  X  3"  X  i"  Zs 

6.9 

4 

■  +  55.300 

lOl" 

8,500 

6.51 

6.9 

8 

4-  52,000 

lOl" 

8,500 

6.12 

6.9 

14 

-|-  48,600 

lOl" 

8,500 

5-72 

6.9 

3—12 

+  6. 700 

50  5" 

.78 

9,000 

■75 

2. 16 

5—  9 

-  7,500 

12,000 

■63 

1.7  net 

7 

+  13,400 

lOl" 

''78 

6,800 

1.97 

2.16 

II 

—  15,000 

12,000 

1-25 

1 . 9  net 

13 

—  22,500 

12,000 

1.88 

1 .9  net 

2 

—  52.500 

12,000 

4.38 

"    4"  X  3"  Zs,  26  lbs.  per  yard 
"    4"  X  3"  Zs,  26  " 
"     3"  X  3"  Zs,  16  " 

4.5  net 

6 

-  45.000 

12,000 

3-75 

4-5  net 

10 

—  30,000 

12,000 

2  .  50 

2.6  net 

denotes  compression;  —  denotes  tension. 


MODERN  FRAMED  STRUCT  URES. 


TABLE  OF  COEFFICIENTS  FOR  CALCULATING  STRESSES  IN  FINK  ROOF  TRUSSES. 


No. 

«  —  3. 

«  —  4. 

«  —  5. 

General  Formulae. 

I 

6.310 

7.826 

9.4247 

+  ^f/«'  +  4 
4 

X  P 

2 

5-25 

7.0 

8.75 

-In 
4 

X  Z' 

3 

0.832 

0.8945 

0.9285 

+  " 

+  4 

X  P 

4 

5-755 

7-379 

9-053 

X  /' 

5 

0.75 

1 .0 

1.25 

n 
4 

X 

6 

4-5 

6.0 

7-5 

2 

X  P 

7 

1.664 

1.789 

1.857 

_|_  2« 

4/«2  -f  4 

X  P 

8 

5.200 

6.932 

8.681 

X  P 

9 

0.75 

I.O 

1.25 

n 

~  4 

X  /' 

lO 

3.0 

4.0 

5.0 

—  « 

X 

II 

1-5 

2.0 

2.5 

I 

 « 

2 

X  P 

12 

0.832 

0.8945 

0.9285 

n 

^  i'V  +  4 

X 

13 

2.25 

3.0 

3-75 

"4" 

X  P 

14 

4.646 

6.485 

8.310 

4/«»  +  4\4  / 

X 

Note.  For  coefficients  for  a  similar  truss  except  that  two  struts  are  used  in  place  of  one  at  3  and  12,  in  the 

figure,  sec  Engineering  News,  Oct.  31,  1895. 


ROOF  TRUSSES. 


319 


It  will  be  noticed  that  the  top  chord  is  made  the  same  area  throughout,  as  it  is  better 
construction  and  costs  very  little  more  than  if  the  chord  were  spliced.  The  members  1 1  and 
13  and  2  and  6  are  made  of  the  same  area  for  the  same  reasons. 

The  detail  sketch  of  this  truss  is  shown  in  Fig  354,  page  320.  The  plates  at  the  joints 
are  all  three  eighths  of  an  inch  thick  except  those  at  the  ends,  which  are  one  half  of  an  inch 
thick.  The  rivets  are  all  three  quarters  of  an  inch  in  diameter.  The  required  number  of 
rivets  through  a  member  connecting  it  at  its  ends  with  a  joint  plate  is  equal  to  the  stress  in 
the  member  divided  by  the  value  of  one  rivet  in  bearing  on  the  joint  plate.  The  number  of 
rivets  connecting  the  bottom  chord  to  the  half-inch  joint  plate  at  the  ends  is  increased 
beyond  what  the  stress  in  the  chord  would  require  owing  to  the  fact  that  the  end  reaction  on 
the  masonry  produces  a  vertical  stress  on  the  rivets  in  addition  to  the  horizontal  stress  from 
the  chord.  The  required  number  of  rivets  for  this  case  is  found  as  follows,  assuming  the  end 
reaction  to  be  resisted  by  the  rivets  directly  over  the  bearing  plate  :  The  end  reaction  is 
26,250  lbs.,  and  as  at  the  ordinary  three-inch  pitch  there  are  six  rivets  resisting  this  end  reac- 
tion, we  have  a  vertical  stress  on  each  rivet  of  26,250  6  =:  4375  lbs.  As  the  value  of  one 
three-quarter-inch  rivet  in  bearing  on  the  half-inch  joint  plate  is  ^  X  |  X  15,000  =  5625  lbs., 
each  rivet  will  resist  -s/ 5625'  ~  4375'  —  3535  lbs.  horizontal  stress  before  the  limiting  bearing 
pressure  is  exceeded.  The  six  rivets  are  then  able  to  resist  21,210  lbs.  of  the  chord  stress, 
leaving  31,040  lbs.  to  be  resisted  by  the  remaining  rivets  through  the  chord  angles.  This 
requires  six  more  rivets,  or  twelve  in  all.  The  splice  at  the  end  of  the  lower  chord  piece  10 
requires  consideration.  There  must,  in  addition  to  sufficient  rivets,  be  enough  area  in  the  splice 
plates  to  equal  the  area  required  in  this  chord.  Of  the  joint  plate,  only  a  width  of  plate  which 
is  symmetrical  about  the  rivet  lines  in  the  chord  angles  is  available  in  the  splice,  and  also  the 
centre  of  gravity  of  the  splicing  plates  should  coincide  with  the  centre  of  gravity  of  the  chord 
angles.  This  latter  condition,  counting  3J  inches  as  the  available  width  of  the  joint  plate, 
requires  6|-  inches  as  the  width  of  the  bottom  splice  plate,  assuming  all  plates  to  be  three 
eighths  of  an  inch  thick.  The  splicing  plates  have  therefore  an  area  equal  to  (3^  -|-  6|— 3X-|)f 
=  2.77  square  inches  net  area,  which  is  sufficient.  If  this  area  had  been  less  than  the  net  area 
required  in  the  chord  10,  the  joint  plate  would  have  to  be  made  thicker. 

It  will  be  noted  that  the  neutral  axes  of  all  the  members  meeting  at  a  joint  intersect  at  a 
point.  It  is  not  more  expensive  to  do  this  in  practice  than  it  is  to  neglect  it,  although  it  is 
not  generally  done.  The  secondary  stresses  in  the  members  due  to  the  fact  that  the  connect- 
ing rivets  at  the  ends  of  the  members  are  not  on  the  neutral  axis,  or  symmetrical  about  it, 
still  remain.  These  stresses  should  be  looked  into  in  order  to  see  whether  there  are  any 
probably  dangerous  stresses.  It  is  not  practicable  to  make  a  rigid  analysis  of  these  stresses, 
owing  to  the  rigid  joints  and  details,  in  the  manufacture  of  which  we  cannot  estimate  the 
effect. 

314.  The  Bracing  of  Roofs  is  usually  put  in  the  plane  of  the  top  chord,  the  purlins 
being  used  as  struts  for  the  lateral  system.  When  a  line  of  shafting  is  supported  near  the 
lower  chord  a  system  of  lateral  bracing  should  be  introduced  there  also,  to  resist  any  vibration 
from  the  shafting.  For  a  building  with  a  number  of  trusses,  bracing  is  not  needed  between 
each  pair  of  trusses,  but  may  be  put  in  at  distances  of  about  100  feet  between  the  bays  which 
have  complete  systems,  the  purlins  being  relied  upon  to  carry  any  side  stress  which  calls  for 
bracing  between  the  necessary  expansion  joints  in  the  purlins.  If  the  end  of  the  building  is 
covered  with  glass,  as  the  end  of  a  train-shed  for  instance,  it  must  be  stiffened  against  bending 
and  the  consequent  destruction  of  the  glass.  This  is  usually  accomplished  by  using  a  hori- 
zontal truss  in  the  plane  of  the  lower  edge  of  the  glass  end  and  another  complete  system  in 
the  plane  of  the  top  chords  of  the  truss.  The  glass  between  these  two  lateral  systems  would 
then  be  stiffened  by  vertical  girders  attached  at  their  ends  to  these  lateral  systems. 


MODERN  FRAMED  STRUCTURES. 


THE  COMPLETE  DESIGN  OF  A  SINGLE  TRACK  RAILWAY  BRIDGE.  321 


CHAPTER  XXI. 

THE  COMPLETE  DESIGN  OF  A  SINGLE  TRACK  RAILWAY  BRIDGE. 

315.  Data. — The  span  length  will  be  150  feet  centre  to  centre  of  end  pins.  Bridge 
square  ended,  not  skew.  Straight  track.  The  capacity  or  live  load  to  be  Cooper's  Class 
"  Extra  Heavy  A"  loading  (^Fig.  129).  The  specifications  for  allowed  stresses,  details  of  con- 
struction, and  material  will  be  the  standard  specifications  of  the  Pennsylvania  Railroad  dated 
1887,  modified  so  as  to  allow  the  use  of  steel  eyebars  and  pins  at  allowed  stresses  20  per  cent 
above  those  specified  for  wrought-iron.  The  panels  may  be  of  any  length  if  the  depth  is 
always  made  equal  to  or  greater  than  the  panel  length,  and  the  stringers  may  be  spaced  any 
distance  centre  to  centre  if  the  size  of  cross-tie  is  made  to  correspond. 

316.  Dimensions  of  Truss. — The  proper  depth  for  economy  in  material  of  a  single  track 
through  span  varies  from  one  fifth  to  one  sixth  of  the  span.  The  depth  is  dependent  to  a 
small  extent  on  the  length  of  panel,  the  economical  depth  being  greater  if  the  panel  is  longer 
and  less  if  the  panel  length  is  made  less.  The  weight  of  the  bridge  varies  very  little  for  a 
change  in  the  number  of  panels,  but  the  cost  of  manufacture  changes  rapidly,  being  less  per 
pound  for  the  longer  panels.  There  are  two  good  reasons  for  selecting  long  panels,  one  being 
a  reduction  in  the  cost  of  the  bridge,  and  the  other  the  concentration  of  a  greater  mass  in  the 
floor  system  to  resist  impact  and  vibration  from  the  train.  The  objection  which  can  be  urged 
against  long  panels  is  tiiat,  owing  to  the  fact  that  the  trusses  for  spans  under  300  feet  are  not 
usually  spaced  much  over  16  feet  centre  to  centre,  the  angle  made  by  diagonals  of  the  lateral 
systems  with  the  centre  line  of  truss  is  so  small  as  to  make  this  bracing  less  efTective  than  if 
the  panels  were  shorter.  This  objection  deserves  consideration,  although  it  is  usually  not 
given  much  weight.  The  greater  number  of  joints  required  for  the  shorter  panel,  and  the 
necessary  play,  or  required  looseness  of  fit,  at  the  joints,  tend  to  counterbalance  this  advan- 
tage. A  panel  length  of  25  feet  and  a  depth  of  28  feet  will  be  assumed,  as  these  dimensions 
are  probably  very  near  the  correct  ones  for  economy  in  material  and  cost.  Five  panels  of  30 
feet  and  a  depth  of  30  feet  may  possibly  be  cheaper,  but  the  panel  length  is  too  long  for  good 
lateral  bracing. 

317.  Dead  Load. — For  the  weight  of  floor  the  specifications  require  that  this  shall  be 
arrived  at  by  adding  to  the  weight  of  the  cross-ties  used  165  lbs.  per  linear  foot  of  track,  which 
includes  the  weight  of  the  rails,  guard  rails,  spikes,  bolts,  etc.  The  cross-tie  will  be  taken  as 
8  inches  by  10  inches,  laid  flat,  and  12  feet  long,  spacing  the  stringers  7  feet  centre  to  centre. 
The  clear  opening  allowed  between  cross-ties  is  6  inches,  and  the  cross-ties  are  therefore  16 
inches  centre  to  centre.  The  weight  of  the  cross-ties  per  linear  foot  of  track  will  be,  assuming 
the  wood  to  weigh  4^  lbs.  per  foot  board  measure  as  specified,  8  X  10  X  12  X  yV  X  i  X  4i 
=  270  lbs.  The  total  weight  of  the  floor  will  therefore  be  270+  165  =  435  lbs.  per  linear 
foot.  The  weight  per  linear  foot  of  the  iron  and  steel  in  the  bridge  will  be  obtained  from  the 
formula  5/4- 350,*  when  rt^  =  weight  per  linear  foot  and  /  =  length  of  span.  The  total 
dead  load  on  the  bridge  will  therefore  be  435  +  iioo  =  1535  lbs.  per  linear  foot.  This  gives 
a  panel  load /^-r  truss  of  19,188  lbs.,  or  for  convenience  19,200  lbs.  We  will  consider  one 
third  of  this  concentrated  at  the  top  chord  joint  and  two  thirds  at  the  bottom  chord  joint. 
This  is  the  usual  division  of  the  panel  load,  but  in  long  spans  it  is  always  best  to  get  the 
correct  division  of  load  from  a  trial  span  worked  out  on  the  basis  of  the  above  division. 


*  For  a  double  track  bridge  add  90^  of  this,  or  for  double  track  w  =  g.5/-|-  665. 


322 


MODERN  FRAMED  STRUCTURES. 


318.  Allowed  Stresses  per  Square  Inch.— We  give  below  an  abstract  of  that  part  of 

tne  specifications  relating  to  the  allowed  stresses. 

The  maximum  and  minimum  stresses  in  compression  and  tension  for  the  specified  loads  are  to  be  used 
in  determining  the  permissible  working  stress  in  each  piece  of  the  structure,  according  to  the  following  for- 
mulae: 

For  pieces  subject  to  one  kind  of  stress  only  (all  compression  or  all  tension), 

a  =  u{i  +r).  [Same  as  eq.  (5),  p.  244.] 

For  pieces  subject  to  stresses  acting  in  opposite  directions, 

a  =  u{i  —  ri).  [Same  as  eq.  (6),  p.  244.] 

In  the  above  formulae, 

a  =  permissible  stress  per  square  inch,  either  tension  or  compression ; 

i  7500  lbs.  per  square  inch  for  double-rolled  iron  in  tension  (eyebars  or  rods), 
« =  <  7000  "     "       "        "     "    rolled  iron  in  tension  (plates  or  shapes), 
( 6500  "     "      "       "     "   rolled  iron  in  compression ; 


r  = 


minimum  stress  in  piece 
maximum  stress  in  piece 

maximum  stress  of  lesser  kind 
2  X  maximum  stress  of  greater  kind' 


The  permissible  stress  a  for  members  in  compression  is  to  be  reduced  in  proportion  to  the  ratio  oi 
the  length  to  the  least  radius  of  gyration  of  the  section,  by  the  following  formula : 

^  =  — ; 


I  + 


18,000^' 

where  a  =  permissible  stress  previously  found  ; 

b  =  allowable  working  stress  per  square  inch  ; 
/  =  length  of  piece  in  inches,  centre  to  centre  of  connections; 
g  =  least  radius  of  gyration  of  section  in  inches. 

The  permissible  stress  in  long  vertical  suspenders  in  through  bridges  to  be  10  per  cent  less  than  that 
given  by  above  formulae. 

The  compression  flanges  of  plate  girders  must  not  have  a  greater  stress  than  that  given  bv  the  following 
formula : 


c  = 


I  + 


5000W' 


where  a  =  permissible  stress  previously  found  ; 

c  —  allowable  working  stress  per  square  inch ; 
/  =  unsupported  length  of  flange  in  inches; 
w  —  width  of  flange  in  inches. 

Iron  pins  are  to  be  so  proportioned  that  the  maximum  fibre  stress  shall  not  exceed  15,000  lbs.  per  square 
inch. 

The  bearing  stress  on  pins,  on  a  diametrical  section  of  pin-holes,  shall  not  be  greater  than  i  J  times  the 
compression  unit  stress  a  in  the  piece  considered. 

The  shear  on  the  net  section  of  any  member  shall  not  exceed  the  compression  unit  stress  a  for  thai 
member,  and  in  case  of  rivets  at  least  20  per  cent  extra  section  must  be  allowed. 

Rivets  must  not  have  a  bearing  pressure  per  square  inch  against  the  web  plates  of  more  than  twice  thf 
compressive  unit  stress  a  used  in  the  upper  flange  of  the  girder. 

The  tension  on  laterals  shall  not  exceed  15,000  lbs.  per  square  inch  for  double  rolled  iron  or  12,000  lb.' 
per  square  inch  for  plates  or  shapes. 


THE  COMPLETE  DESIGN  OF  A  SINGLE  TRACK  RAILWAY  BRIDGE,  3aj 


The  compressive  stress  on  lateral  struts  shall  not  exceed  that  given  by  the  following  formula : 

12,000 


1  + 


eg" 


where  d  =  permissible  stress  per  square  inch  ; 

/=  length  of  piece  centre  to  centre  of  connections  in  inches; 
g  =  least  radius  of  gyration  of  the  piece  in  inches ; 

{36,000  for  both  ends  fixed ; 
24,000  for  one  end  hinged,  the  other  fixed ; 
18,000  for  both  ends  hinged. 
[The  student  will  bear  in  mind  that  for  steel  eyebars  and  pins  the  above  allowed  stresses  will  be  increased 
20  per  cent.] 

Minimum  Sections. — No  iron  shall  be  used  of  less  thickness  than  three  eighths  of  an  inch,  except  in 
lateral  struts,  lattice  straps,  pin-plates,  and  similar  details,  and  in  special  cases  of  girder  flanges,  where  a 
minimum  thickness  of  five  sixteenths  will  be  allowed. 

No  iron  used  in  compression  members  shall  have  an  unsupported  width  of  more  than  forty  times  its 
thickness. 

No  rod  shall  be  used  having  a  less  sectional  area  than  one  square  inch. 

319.  Tabulation  of  the  Stresses  and  the  Determination  of  the  Sectional  Areas  of 
the  Members  of  the  Truss. — It  is  assumed  that  the  student  can  calculate  the  stresses  in 
the  truss  members  from  the  dead  and  live  loads  by  the  methods  explained  in  Part  I.  In  the 
following  table  are  given  the  stresses  on  the  various  pieces,  with  the  factors  which  determine  the 

B 


Fig.  355. 


Stresses. 

Mem- 
:hes. 

% 

a 

"0  d 

—  c 

«.2 

Unit 
Stress. 

Area 

requirec 

Make-up  of  Section. 

Area 

Material. 

u 

r-  C 

■J  « 

-0  u 

1. 

Used. 

Dead. 

Live. 

Mini- 
mum. 

Maxi- 
mum. 

.0 
b 

Lengti 
ber  ii 

fl  >. 
DSO 

ab 

-  42.9 

—  iir.o 

42 

9 

153-9 

II. 5 

It. 5 

13-4 

Two  6"  X  li"  bars 
Two  6"  X  i|"  bars 

13-5 

Steel 

be 

-  42.9 

—  Ill  .0 

42 

9 

153-9 

II-5 

"•5 

13.4 

13-5 

<( 

cd 

—  68.6 

-169.5 

68 

6 

238.1 

II. 6 

II. 6 

20.5 

Two  6"  X  if"  bars 

21 .0 

-3^-6 
—  12.9* 

(+  8.4 

Be 

,( —  no.  8 

30 

2 

149.4 

10.8 

10.8 

13.8 

Two  5"  X  if  bars 
Two  5"  X   r  bars 

13-75 

(( 

Cd 

-  6!t>9 

0 

0 

77-8 

9.0 

9.0 

8.7 

8.75 

i( 

De 

-  17-5 

0 

0 

17-5 

7-5 

7-5 

2-3 

One  ly  square  rod 
Two  4"  X  If"  bars 

2.25 

Iron 

Bb 

—  12.8 

—  59-4 

12 

8 

72.2 

10.6 

9-5 

7.6 

7-5 

Steel 

Cc 

-pi6.o 

+  48.4 

16 

0 

64.4 

8.1 

336 

3-8 

5-64 

II. 4 

Two  10"  i_is.  58  lbs.  per  yd. 

II. 6 

Iron 

Dd 

+  6.4 

+  I3-I 

6 

4 

19-5 

8.6 

336 

2-54 

4-360 

4-5 

Two   7"  I— IS,  38  lbs.  per  yd. 
One  22"  X  xff"  top  plate 

• 

7.6* 

aB 

+  64^4 

+166.6 

64 

4 

231.0 

8-3 

450-5 

6-34 

6.460 

35-5  ■ 

Two   3"  X  3"/s,  30  lbs.  per  yd. 
Two  16"  X  rff"  side  plates 
Two  3"  X  4"^s,  30  lbs.  per  yd. 
One  22"  X        top  plate 

■35-6 

Iron 

BC 

+  68.6 

+169.5 

68 

6 

238.1 

8.4 

300 

6.44 

7.470 

31.9  ^ 

Two  3"  X  3"  X  1"  Zs 

Two  16"  X  1"  side  plates 

Two   3"  X  4"Zs,  30  lbs.  per  yd. 

►  31.9 

Iron 

One  22"  X  VV   top  plate 

CD 

+  77-1 

+192.9 

77 

I 

270.0 

8.4 

300 

6.44 

7.470 

36.2  - 

Two  3"  X  3"Zs,  30  lbs.  per  yd. 

Two  16"  X  iV"  side  plates 

Two  3"  X  4"  Zs,  33  lbs.  per  yd. 

■36.2 

Iron 

Areas  are  given  in  square  inches.    Stresses  are  given  in  thousand-pound  units, 
denotes  compression.    —  denotes  tension. 


The  area  given  to  this  post  by  the  formuls  is  altogether  too  small  for  lateral  stability.  ; 

■A 

I 
i 


324 


MODERN  FRAMED  STRUCTURES. 


allowed  stresses,  the  required  areas,  the  make  up  of  the  section,  and  the  area  used,  from  which 
the  student  can  readily  understand  the  methods  employed.  The  members  are  designated  by 
the  letters  of  the  joints  at  their  ends  as  indicated  in  Fig.  355. 

The  order  in  which  the  members  are  arranged  in  this  table  is  the  most  convenient  one  to 
use  when  making  up  the  sections.  The  sizes  of  the  tension  members  are  fixed  first,  then  the 
sizes  of  the  posts,  and  lastly  the  inclined  end  posts  and  top  chords,  which  are  usually  made 
with  the  same  general  dimensions.  The  top  chords  must  be  made  wide  enough  to  allow  the 
posts  and  diagonals  to  be  packed  inside  of  them,  and  they  must  also  be  made  deep  so  that 
the  neutral  axis  or  pin  centre  may  be  far  enough  from  the  top  plate  to  allow  room  enough  for 
the  eyebar  head. 

Owing  to  the  requirement  of  established  practice  that  posts  must  never  have  a  ratio  of 
length  to  least  width  of  over  48,  the  posts  Dd  are  made  heavier  than  the  stresses 
would  require.  A  lighter  7-inch  channel  could  have  been  used  except  for  the  clause  in  the 
specifications  allowing  no  metal  less  than  three  eighths  of  an  inch  thick. 

The  external  dimensions  of  the  top  chord  section  were  determined  as  follows :  The 
minimum  width  of  top  plate  is  fixed  by  the  members  which  have  to  be  packed  inside  at  joint 
C,  where  the  largest  post  is.  The  side  plate  is  determined  by  the  piece  BC,  where  the 
minimum  section  is  required,  as  this  area  must  be  made  up  without  using  material  of  less 
thickness  than  the  specifications  allow.  By  referring  to  Fig.  356,  it  will  be  seen  that  22  inches 
is  the  least  allowable  width  for  the  top  plate,  requiring  a  plate  at  least  seven  sixteenths  of  an 
inch  thick  in  order  that  the  ratio  of  the  unsupported  width  to  the  thickness  of  the  plate  may 
not  exceed  40.  The  unsupported  width  is  the  distance  between  the  rivets  which  attach 
the  plate  to  the  top  angles,  and  is  usually  4  inches  less  than  the  width  of  the  plate.  The 
depth  of  the  chord  or  the  width  of  the  side  plates  must  be  determined  by  trial.  The  least 
radius  of  gyration  of  a  top  chord  section  does  not  vary  much  from  four  tenths  of  the  width  of 
the  side  plate  when  the  area  of  the  side  plates  is  one  half,  or  less  than  one  half,  of  the  total 
area  of  the  section.  Using  this  as  a  basis  for  an  approximate  determination  of  the  area  in 
BC  iox  various  widths  of  side  plate,  we  select  the  section  with  a  width  of  side  plate  which 
will  give  us  the  least  metal  in  the  top  chords  and  end  posts.  For  short  spans  this  is  generally 
the  one  which  will  give  the  required  area  in  BC  when  the  minimum  attowed  sections  are 
used  in  its  make-up.  The  angles  used  in  these  sections  are  as  small  as  should  be  used  for 
convenience  in  driving  rivets.  Care  should  be  taken  to  select  a  chord  section  deep  enough 
to  allow  the  pin  to  be  placed  at  such  a  distance  from  the  top  plate  that  the  eyebar  head  will 
not  strike  the  top  plate.    The  pin  is  usually  placed  below  the  neutral  axis  of  the  section  in 

order  that  the  total  direct  stress  acting  with  a  lever  arm 
equal  to  the  distance  of  the  centre  of  the  pin  from  the 
neutral  axis  may  produce  a  moment  equal  to  the  centre 
bending  moment  in  the  piece  from  its  weight.  As  the 
jjj^,.  pins  are  at  the  same  distance  from  the  top  plate  in  all 
the  sections  of  top  chord  and  end  post,  this  requirement 
must  be  observed  in  the  design  of  the  sections  in  order 
to  avoid  any  unnecessary  bending  stresses.  The  pin 
should  never  be  placed  at  such  a  distance  from  the 
neutral  axis  that  the  algebraic  sum  of  the  bending  mo- 
ments produced  by  the  maximum  direct  stress  and  the 
356.  weight  of  the  member  would  cause  an  extreme  fibre 

stress  per  square  inch  greater  than  10  per  cent  of  the  stress  per  square  inch  allowed  by  the 
formula  for  direct  stress  alone. 

The  top  chords  and  end  posts  for  this  span  are  made  of  the  general  dimensions  shown  in 
Fig.  356.    It  will  be  noted  that  the  vertical  distance  back  to  back  of  angles  is  one  quarter 


— 


THE  COMPLETE  DESIGN  OF  A  SINGLE  TRACK  RAILWAY  BRIDGE.  325 


of  an  inch  more  than  the  width  of  the  side  plate.  This  is  required  in  manufacture.  The 
pin  centre  will  be  placed  if  inches  above  the  centre  of  the  web  plate.  In  the  following 
table  are  given  the  distances  of  the  neutral  axes  from  the  centre  of  the  web  plates,  the  weights 
of  the  pieces,  the  maximum  bending  moments  produced  by  the  weights  of  the  pieces,  con- 
sidering each  piece  as  a  beam  of  a  span  equal  to  the  length  of  the  piece,  the  maximum  direct 
stresses  in  each  piece  from  the  dead  and  live  loads,  and  the  distance  the  pin  must  be  placed 
from  the  centre  of  the  plate  in  order  that  the  direct  stress  may  produce  a  reverse  bending 
moment  which  will  counteract  the  moment  of  the  weight.* 


Piece. 

e 

W 

S 

S 

aB 

2.23 

5,100 

igi,200 

231,000 

0.83 

1 .40 

BC 

2.  II 

3,000 

1 12,500 

238,100 

.48 

I  .63 

CD 

2.08 

3.400 

127,500 

270,000 

•47 

I. 51 

In  this  table  e  is  the  eccentricity  or  distance  from  the  centre  of  the  web  plate  to  the  neutral  axis;  /Fis 
the  weight  of  the  member,  allowing  fifteen  per  cent  of  the  area  as  the  weight  of  the  rivet-heads,  lattice  bars, 
etc.;  is  the  centre  bending  moment  produced  by  the  weight  W ;  .S'  is  the  direct  stress  in  the  member 
assumed  to  be  applied  at  the  centre  of  the  end  pins  of  the  piece  ;  and  d  is  the  distance  the  pin  centre  should 
be  placed  from  the  centre  of  the  web  plate  in  order  that  the  bending  moment  produced  by  the  weight  of  the 
piece  may  be  counteracted  by  the  moment  of  the  direct  stress  acting  about  the  neutral  axis  of  the  piece. 

As  any  future  increase  of  the  live  load  would  cause  greater  direct  stress  in  the  piece,  and 
consequently     would  be  larger,  it  is  better  to  use  one  of  the  larger  values  of  r,. 

The  method  of  computing  the  eccentricity  e  and  the  radius  of  gyration  is  given  in  the  following  tables  : 

c 


ECCENTRICITY  FOR  aB. 


Piece. 

Area,  A 

Distance,  d 

A  y.d 

e  =  8.34  -  rf, 

Top  angles.  .  .  . 
Web  plates.  .  .  . 
Bottom  angles. 

Total  section . . 

9.625 
6.0 
14.0 
6.0 

0.0 
1. 16 
8.34 
15-65 

6.96 
116.76 
93-90 

35-625 

6. II 

217.62 

2.23 

I 


3 


— B 


o 

I 

Fig.  357. 

=  distance  of  the  centre  of  gravity  of  the  piece  from  the  centre  of  the  top  plate  ;  di  =  distance  of  the 
centre  of  gravity  of  the  total  .section  from  the  centre  of  the  top  plate;  c  =  the  eccentricity,  or  the  distance 
of  the  centre  of  gravity  of  the  total  section  from  the  centre  of  the  web  plate  ;  8.34  =  the  distance  from  the 
centre  of  the  web  plate  to  the  centre  of  the  top  plate.    All  distances  in  inches,  and  all  areas  in  square  inches, 


THE  RADIUS  OF  GYRATION  FOR  aB.\ 


Piece. 

Area,  A 

C 

/J  X  <r» 

I 

"  1       1  1  —1  —  MM 

A  X      -f  / 

£ 

Top  plate  

Bottom  angles. . 
Total  section . .  . 

9.625 
6.00 
14.00 
6.00 

6.11 

4-95 
2.23 

9-54 

37-33 
24.50 
4-97 
91.01 

359-30 
147.00 
69.58 
546.06 

ai5 
4.72 
298.67 
4-44 

359-45 
151.72 
368.25 
550.50 

35  625 

0 

40.14 

1429.92 

6-34 

*  It  is  here  assumed  that  the  total  chord  stress  passes  through  the  pin.    If  only  the  horizontal  component  of  the 
stress  in  the  diagonal  comes  to  the  pin  it  is  impracticable  to  fully  counteract  the  dead  weight  of  the  chord  section, 
t  See  Art.  126  for  another  form  of  tabular  computation  of  /  and  r. 


326 


MODERN  FRAMED  STRUCTURES. 


A  is  the  area  of  each  piece  in  square  inches ;  c  is  the  distance  of  the  centre  of  gravity  from  th«^  neutral 
axis  of  the  full  section  :  f  is  the  moment  of  inertia  of  each  piece  with  reference  to  the  neutral  axis  of  the 
piece  parallel  to  AB  (see  Fig.  357; ;  and^  is  the  radius  of  gyration  of  the  full  section  of  aB.  All  areas  are 
in  square  inches,  and  all  distances  or  dimensions  in  inches. 

The  position  of  the  neutral  axes  and  the  moments  of  inertia  for  the  angles  are  obtained  from  any  of  the 
handbooks  issued  by  the  rolling  mills. 

The  above  computations  are  made  with  reference  to  an  axis  parallel  to  AB  (Fig.  357).  The  radius  of 
gyration  with  reference  to  the  axis  CZ?  should  alv/ays  be  equal  to  or  greater  than  that  found  above.  This  i«true 
whenever  the  width  between  the  outer  sides  of  the  side  plates  is  nine  tenths  of  the  breadth  of  the  side  plates. 

320.  The  Design  of  the  Iron  Floor  System. — Having  fixed  the  size  of  the  truss 
members,  we  will  now  take  up  the  floor-beams  and  stringers.  The  stringers  are  plate  girders 
of  a  span  length  equal  to  the  panel  length.  The  dead  load  on  a  pair  of  stringers  consists  of 
the  weight  of  the  stringer  plus  the  weight  of  the  floor.  The  floor  we  have  found  weighs  435 
lbs.  per  foot  of  track.  The  iron  weight  may  be  approximated  by  the  formula  9  X  -^H-  55. 
where  /  =  length  of  panel.  This  is  the  formula  previously  used  for  deck  plate-girder  spans 
with  the  constant  1 10  reduced  one  half  to  allow  for  the  weight  of  the  bracing,  which  is  not 
generally  used  for  stringers  under  30  feet  long.  The  total  dead  load  per  linear  foot  on  a 
pair  of  stringers  is,  therefore,  435  -|-  280  (=  9  X  25  -f-  55)  =  715  lbs.    The  maximum  bending 

715X25 

moments  for  one  stringer  are  r—  =  27,900  ft.-lbs.  from  the  dead  load  and  234,700  ft.-lbs. 

2X0 

for  the  live  load.  The  live  load  moment  is  taken  from  the  table  on  page  329.  The  total  or 
maximum  bending  moment  is  262,600  ft.-lbs.  The  economical  depth  for  the  stringers  v  ill  be 
found  by  the  formula  given  in  Chapter  XIX  for  a  girder  span  whose  flanges  are  of  constant 

/  in 

area  end  to  end.    In  this  case  h  =  1.41-^/  y^,  where  //  =  the  depth  of  girder,  m  =  the  bending 

moment  in  inch-pounds, /"  =  the  average  flange  stress  per  square  inch  on  the  gross  area  of 
flange,  and  /  =  the  thickness  of  web  plate  in  inches.  In  the  case  of  the  stringers  under  con- 
sideration /  —  three  eighths, /"=  6600,  and  m  =  3,151,200;  from  which  h  =  50.  The  specifi- 
tions  require  a  cover  plate  on  the  top  flange  from  end  to  end  of  the  stringer,  and  as  we  will 
assume  our  stringers  to  be  48  inches  back  to  back  of  flange  angles,  the  distance  centre  to 
centre  of  gravity  of  the  flanges  will  be  taken  as  46^^  inches.  The  maximum  flange  stress 
will  be  3,151,200 46^  =:  67,800  lbs.    The  allowed  stress  per  square  inch  on  the  bottom  is 

/    ,    27,900  \  „       ^,  .    ,  r  ,  ^         .  67,800 

7000^1  +  262  600/  ~  775*^  ^"^^  required  net  area  of  bottom  flange  is  ■  yy^^  =  ^-77 

square  inches.  Two  6"  X  4"  angle  irons,  weighing  49  lbs.  per  yard  each,  will  be  used  for  this 
flange.    The  allowed  stress  per  square  inch  for  the  top  flange  is  6500^1  -(-  ^^^~)  =  7206  lbs., 

7200 

reduced  by  the  formula  —  ^, — ,  where  /  =  25  and  d  =  i.    From  this  we  get  6400  as  the 

1  + 


5000^' 

allowed  stress  per  square  inch  in  the  top  flange.  The  required  area  of  top  flange  is 
67,800  6400  =  10.6.  For  this  flange  use  two  5"  X  32"  angle  irons,  weighing  31  lbs.  per 
yard,  and  one  I2  X  f  inch  plate,  making  a  gross  area  of  10.7  square  inches.  For  stiffeners 
we  will  use  3"  X  2"  X  angles  except  at  the  ends.  The  end  stiffeners  have  to  be 
large  enough  to  allow  f"  rivets  to  be  driven  through  each  leg  of  the  angle.  This  limits 
them  to  3"  X  3"  angles  as  a  minimum,  and  in  order  to  make  the  driving  of  the  field 
rivets  less  confined  we  will  use  3^"  X  3i"  X  |"  angles  for  the  end  stiffeners.  The  end 
shear  is  4500  lbs.  from  the  dead  1^)ad  and  43,700  lbs.  from  the  live  load,  making  a 
total  of  48,200  lbs.  This  requires  11  rivets  through  the  end  stiffeners  and  the  web  of  the 
stringer.    The  value  of  a       rivet  in  bearing  on  a  |-inch  web  plate  is,  as  the  allowed 


THE  COMPLETE  DESIGN  OF  A  SINGLE  TRACK  RAILWAY  BRIDGE.  327 


bearing  pressure  per  square  inch  is  twice  the  allowed  stress  per  square  inch  in  the  top 
flange  before  reduction  for  the  length  of  flange,  2  X  7200  X  |  X  |  =  4700  lbs.  The  nunnber 
of  rivets  through  these  end  stiffeners  and  the  web  of  the  floor-beam  is  determined  as  follows  : 
the  maximum  end  shear  of  the  stringer,  in  this  case  48,400  lbs.,  must  be  transferred  to  the 
floor-beam  by  the  rivets  in  single  shear,  or  the  maximum  concentration  on  the  floor-beam  from 
the  two  stringers  which  attach  on  opposite  sides  of  the  beam  must  be  transferred  to  the  floor- 
beam  by  the  rivets  without  exceeding  the  limiting  bearing  pressure  allowed  between  the 
rivets  and  the  web  of  the  beam.  The  allowed  bearing  pressure  per  square  inch  for  the  rivets 
on  the  web  of  the  beam  is,  as  the  dead  load  concentration  is  9000  lbs.  and  the  maximum  live 

/  QOOO  \ 

load  concentration  is,  from  the  table  page  329,  59,500  lbs.,  2  X  6500^1  +  68^00)  ~  ^^^1^  't)s., 

and  the  bearing  value  of  one  f-inch  rivet  on  a  f-inch  plate  is  f  X  1  X  Hijoo  =  4800  lbs. 
The  maximum  end  shear  of  one  stringer  requires,  as  the  value  of  a  f  rivet  in  single  shear  is 
3660  lbs.,  48,400  3660  =13  rivets.  The  maximum  concentration  on  the  floor-beam 
requires,  as  the  value  of  one  \  rivet  on  the  f-inch  web  of  the  beam  is  4800  lbs.,  68,500  4800 
=  15  rivets.  The  rivets  through  the  flange  angles  and  web  plate  would  be  spaced  by  the 
methods  explained  in  Chapter  XIX. 

321.  Floor-beams. — The  length,  centre  to  centre,  of  end  supports  of  a  floor-beam  should  be 
taken  as  the  distance  centre  to  centre  of  trusses,  and  provision  should  be  made  for  connecting 
the  beam  so  that  its  end  reaction  will  be  transferred  to  the  truss  as  nearly  central  as  practicable. 

The  weight  of  the  beam  can  be  arrived  at  by  trial.  The  depth  of  the  beam  must  usually 
be  great  enough  to  allow  the  full  depth  of  the  stringer  and  about  one  half  inch  of  clearance 
between  the  vertical  legs  of  the  beam  flange  angles,  or  in  this  case  48"  +  k"  +  7"  =  55i"- 
If  the  stringers  are  made  the  economical  depth,  this  usually  requires  the  beams  to  be  deeper 
than  would  be  economical. 

The  above  applies  to  stringers  framed  in  between  the  floor-beams.  When  the  stringers  are  supported 
on  top  of  the  beams  both  may  be  made  the  economical  depth.  The  depths  required  by  the  formula;  for 
economy,  like  the  results  of  all  formulae  for  maximum  or  minimum,  may  be  deviated  from  considerably 
without  much  loss  of  weight. 

Assuming  the  weight  of  the  beam  as  2400  lbs.,  we  get,  remembering  that  the  stringers  are 
7  feet  and  the  trusses  16  feet  centre  to  centre,  ^^oo  X      _         ft. -lbs.  centre  moment  from 

o 

the  weight  of  the  beam,  9000  X  4a  =  40,500  ft. -lbs.  centre  moment  from  the  concentrated 
dead  load  of  the  stringers  on  the  beam,  and  59,500  X  4f  =  267,750  ft.-lbs  centre  moment 
from  the  live  load  concentrated  at  the  stringer  points.  It  will  be  noted  that  the  moments 
from  the  stringer  concentrations  are  constant  between  stringer  points.  The  total  centre 
moment  is  therefore  45,300  ft.-lbs.  from  the  dead  load  and  267,750  ft.-lbs.  from  live  load,  or, 
combining  them,  313,050  ft.-lbs.  As  the  beam  mu.st  be  55I"  deep  and  the  specifications 
require  a  top  flange  plate,  we  will  assume  54  inches  or  4.5  feet  as  the  effective  depth.  The 
flange  stress  is  therefore  313,050  -f-  4.5  =  69,600  lbs.    The  allowed  stress  per  square  inch  on 

the  bottom  flange  is  7000(1  -f  ^^^^^)  =  8000  lbs.,  requiring  a  net  area  of  bottom  flange  of 
69,600     8000  =  8.7  square  inches.    Use  two  6"  X  4"  angles  weighing  49  lbs.  per  yard.  The 
allowed  stress  per  square  inch  on  the  top  flange  area  is  6500(1  -(-  ^'^'^^  j  —  7440  lbs.,  reduced 
7440 

by  the  formula  —  =  7366  lbs.,  as  /  =  7  feet  and  b  =  i  foot.    The  required  area  of 

top  flange  is  69,600  ^  7366  =  9.45  square  inches.  Use  two  5"  X  3"  X  |"  angles  and  one 
12  X  tV  plate. 


328 


MODERN  FRAMED  STRUCTURES. 


The  allowed  shearing  stress  per  square  inch  on  rivets  in  the  floor-beam  is,  according  to 
the  specifications,  7440  -r-  1.2  =  6200  lbs.  The  value  of  one  \  rivet  in  single  shear  is  therefore 
.61  X  6200  =  3782  lbs.  The  allowed  bearing  stress  per  square  inch  for  rivets  on  the  web  of 
the  floor-beam  is  2  X  7440  =  14,880.  The  value  of  one  \  rivet  bearing  against  a  f  plate  is 
^  X  I  X  14,880  =  4880. 

The  number  of  rivets  required  through  the  web  of  the  beam  and  the  end  angles  which 
rivet  to  the  post  of  the  truss  is  69,700  4880  —  15.  The  number  of  rivets  needed  through 
these  end  angles  and  the  post  is  69,700  3782  =:  19.  The  pitch  of  the  rivets  through  the 
flanges  and  web  plate  is  constant  between  the  stringer  point  and  the  end  of  the  beam  and  is 

4880   X  52 

— =  3.64  inches.    Between  the  stringer  points  the  flange  stress  is  practically  constant, 

so  that  6-inch  spacing,  the  maximum  allowable,  will  be  used. 

322.  The  Design  of  the  Lateral  Bracing. — The  wind  pressure  specified  is  30  lbs.  per 
square  foot,  counting  as  the  exposed  surface  twice  the  area  of  one  truss  as  seen  in  elevation 
and  the  surface  of  the  floor.  This  gives  130  lbs.  per  linear  foot  as  the  wind  force  for  the  top 
lateral  bracing,  and  230  lbs.  per  linear  foot  for  the  wind  force  for  the  bottom  lateral  bracing. 
The  train  is  assumed  to  expose  a  surface  10  feet  in  height,  or,  as  it  is  usually  taken,  the  moving 
wind  load  is  300  lbs.  per  linear  foot.  From  the  foregoing  we  get  3250  lbs.  as  the  panel  load 
for  the  upper  lateral  system,  and  5750  lbs.  as  the  static  and  7500  lbs.  the  moving  panel  loads 
for  the  lower  lateral  system.  The  stresses  in  the  lateral  systems  with  the  sections  used  are 
given  in  the  following  tables. 

BCD 

rr 


Piece. 

Total  Stress. 

/ 

g 

/ 

Ar 

Make-up  of  Section. 

Area. 

Material. 

DD' 
CC 
CD 
BC 

1 ,62  = 
3.250 
3.014 
9,043 

192" 
192 

1.19 
1. 19 

7.000 
7.000 
1 5 . 000 
1 5 . 000 

0.23 
0.47 
0  20 

0.  60 

Four  2\"  X  ih"  X^"  Zs 
Four  2i"  X  2i"  X  tV  Zs 
One  i"  sq.  rod 
One  l"  sq.  rod 

6.0 
6.0 
1.0 

1 .6 

Iron 
<< 

/  =:  length  of  piece  in  inches;  ^  =  least  radius  of  gyration  of  piece  in  inches;  =  allowed  stress  per  square  inch; 
Ar  =  area  required. 

a  b  c  d  c  b 


Piece. 

Total  Stress. 

/ 

S 

/ 

Ar 

3-4 

aa' 

hb' 

cc' 

dd' 

ab' 

be' 

cd' 

33,100 
1 9, 600 
8,600 
4,750 
61,400 
38,300 
J  7,900 

84 

0.94 

9,900 

15,000 
15,000 
15,000 

4  I 
2.55 
1 .2 

Make-up  of  Section. 


Two  3"  X  3"  X  A"  Zs 

Lower  flange  of  the  floor- 
beam  used  for  these  struts 

Two  lys"  sq.  rods 
Two  li  '  sq.  rods 
One  iV  sq.  rod 


Area. 


3-6 


4. 12 

2-53 
1.27 


Material. 


Iron 


Iron 


The  strut  aa  is  attached  to  the  end  post  and  to  the  stringers;  the  least  unsupported  length  is  between  the  stringers 
and  is  seven  feet. 


THE  COMPLETE  DESIGIV  OF  A  SINGLE  TRACK  RAILWAY  BRIDGE.  329 

MAXIMUM  SHEARS  AND  BENDING  M3MENTS  ON  GIRDERS  FOR  COOPER'S  CLASS  EXTRA 

HEAVY  "A"  LOADING. 


DC 

pc 

pc] 

PI 

P  1 

)  C 

:)(/ 

?0 

pc 

—871 — > 

♦  -5:8^ 

<^;'8> 

'r-5:8-» 

<-4.'8-' 

< — 8.'2— > 

■<— 8^1 — » 

«-5.'8-» 

♦4.'5-<- 

<— 771— ► 

■^4'8'f-5.'8-> 

M.'8-' 

'-4.'0> 

Wclglits  given  arc  for  one  Rail. 


TjC  n^l  h 

Shear 

Shear 

^laxiroum 

Length 

Shear 

Shear  at 

Shear 

N'T  3x  1  m  u  ro 

'of 

at 

Quarter 

at 

Moment  near 

at 

Quarter 

at 

Moment  near 

Girder. 

End. 

Point. 

Centre. 

Centre. 

Girder. 

End. 

Point. 

Centre. 

Centre. 

10'  0" 

24.8 

15-8 

8.3 

45.0 

43'  0  " 
44'  0" 
45'  0  ' 
46'  0" 

60.4 

37-1 

16.7 

580.3 

11'  0" 

26.6 

16.4 

8.9 

56.3 

61.4 

37-6 

17.0 

602. 1 

12'  0'' 

28.1 

16.9 

9-4 

67.5 

62.2 

38.2 

17.3 

625.8 

13'  0" 

29.4 

18.2 

9.8 

78.8 

63. 1 

38.7 

17.6 

649-5 

14'  0" 

30.5 

19.3 

10.2 

90.0 

47'  0" 

64.0 

39-2 

17.8 

673-3 

15'  0" 

31.7 

20.2 

10.3 

IOI.3 

48'  0" 

64.8 

39-7 

18.I 

697.0 

16  0" 

33-5 

21. 1 

10.3 

112. 5 

49;  0" 

65.7 

40.2 

18.3 

720.7 

17'  0" 

35-0 

2r.8 

10.3 

123.8 

50'  0  " 

51'  0" 

66.5»* 

40.7 

18.6 

744-4 

18'  0" 

3f'-4 

22. 5 

10.3 

135-0 

67.4 

41-3 

18.8 

768.2 

19'  0" 

37-7 

23.1 

10.7 

146.3 

52'  0" 

68.2 

41.8 

19.0 

703.4 

20'  0" 

38.8 

23.8 

II. I 

160.4 

53'  0" 
54'  0" 

6g.  I 

42.4 

19.3 

819.3 

21'  0" 

39  8 

24.8 

II. 4 

175.2 

69.9 

42.9 

19-5 

845.2 

22'  0" 

40.7 

25-7 

II.  7 

189.9 

55'  0" 

70.8 

43-4 

19.8 

871. 1 

23'  0" 

41.6 

26.5 

12.0 

204.7 

56'  0" 
57'  0  " 

71.6 

43-9 

20.1 

897.4 

24'  0" 

42.7 

273 

12.2 

219  6 

72.4 

44-3 

20.3 

925.6 

25'  0" 

43-7 

28.0 

12.2 

234-7 

58'  0" 

73-2 

44-8 

20.6 

953-9 

26'  0" 
27'  0" 

44.6 

28.7 

12.3 

251-9 

59'  0" 

74-1 

45  2 

20.8 

982.1 

45-5 

29- 3 

12.3 

269. 1 

60'  0" 

74-9 

45-8 

21. 1 

1010.3 

28'  0" 

4&-3 

29.8 

12.4 

286.4 

61'  0 " 

75-7 

46.2 

21.3 

1038.6 

29'  0" 

47.0 

30.3 

12.4 

303.6 

62'  0" 

76.5 

46.7 

21.5 

1066.8 

30'  0" 

48.0 

30.7 

12.7 

320.9 

63'  0" 

77-3 

47-2 

21.7 

1095.2 

31'  0" 

48.9 

•^1.2 

13.1 

338.1 

64' 

65'  0" 
66'  0" 

78.0 

47.6 

21.9 

•125.7 

32'  0" 

49.8 

31.8 

13-5 

355-3 

79-1 

48.1 

22.1 

1 1 56.2 

33'  0" 

50.8 

32.3 

13-9 

373-7 

80.1 

48  6 

22.4 

1187.7 

34'  0" 

51.9 

32.8 

14-3 

393- 1 

67'  0 " 

81. 1 

49.0 

22.7 

1219. 1 

35'  0" 

52.9 

33-2 

14.6 

4 1 2.6 

68'  0  " 

92.1 

49.4 

22.9 

1250.6 

36'  0" 

53.8 

33-6 

14.9 

432.1 

69'  0'' 

83.2 

49-8 

23. 1 

1282.4 

37'  0" 

54-7 

34.1 

15  2 

451-6 

70'  0" 

84.4 

50.3 

23-4 

1314-3 

38'  0" 

55.8 

34-4 

15-5 

472  7 

71'  0" 

85.5 

50.7 

23.6 

•346.4 

39'  0" 

56-8 

35-' 

15.8 

494-3 

72'  0" 

86.6 

51-I 

23.8 

1378.6 

40'  0" 

57-8 

35-6 

16.0 

515-8 

73'  0" 
74'  0" 
75'  0" 

87-7 

52.0 

24.0 

141 1. 0 

41'  0" 

58.7 

36. 1 

lO  3 

537-3 

88.7 

52.1 

24.2 

•443-8 

42'  0" 

59-6 

36.6 

16.5 

558.8 

89.7 

52.4 

24.5 

1476-7 

323.  The  Design  of  the  Portal  Strut.* — The  wind  force  concentrated  at  the  hip  joint 

is  assumed  to  be  carried  to  the  abutments  by  the  incUned  end   

posts.  These  posts  act  as  beams  fixed  at  both  ends,  and  the 
distribution  of  external  forces  will  be  as  shown  in  Fig.  360.  The 

Ph 

maximum  bending  moment  in  the  portal  strut  is  — ,  and  this  is 

4 

also  the  maximum  bending  moment  in  the  end  posts.  The  as- 
sumption that  the  end  posts  are  fixed  at  their  lower  ends  is  de- 
pendent upon  the  amount  of  direct  stress  in  the  end  post  and 
the  width  of  the  end  posts.  It  is  true  whenever  the  total  stress 
in  the  end  post  multipHed  by  one  half  the  distance  centre  to 

Ph 

centre  of  bearings  on  the  end  pin  is  equal  to  or  greater  than  — . 

4 

The  force  P,  in  the  case  of  the  150-foot  span  under  consider- 
ation, is  eoual  to  2^  X  3250  =  8125  Ib.s.;  h  =  450.5  inches  and 
d  —  \  js.    The  portal  strut,  in  order  to  give  the  specified 

clearance  nom  the  rail  to  the  under  side  of  the  strut  of  20  feet,  is 
about  30  inches  deep.    It  will  be  assumed  to  be  29  inches  centre  to  centre  of  gravity  of  its 
*  See  Arts.  115,  151,  atid  "  Elevated  Railways  "  in  Chapter  XXV. 


33° 


MODERN  FRAMED  STRUCTURES. 


flanges.   The  maximum  stress  in  each  flange  of  the  portal  in  order  to  fix  the  posts  in  direction 

at  the  top,  will  then  be  ^^"^  ^  450-5  ^  _L  _  31,600  lbs.    To  this  stress  must  be  added  —  a 

4  29  4 ' 

direct  compression  due  to  the  direct  application  of  the  wind  force  at  the  end  of  the  strut,  one 
half  of  the  force  P  being  assumed  as  resisted  by  each  end  post.  The  total  stress  on  each  flange 
of  the  portal  strut  is,  therefore,  31,600  +  2000  —  33,600.  Assuming  each  flange  to  be  com- 
posed of  two  3"  X  32-"  angles,  the  allowed  stress  in  compression  is  found  to  be  8700  lbs.  per 
square  inch,  requiring  3^^  square  inches  area.  Two  3"  X  3^"  X  jV"  angles  are  therefore 
sufificient.  The  portal  strut  is  usually  made  in  the  form  of  a  girder,  the  web  being  either 
lattice  work  or  a  solid  plate.  The  latter  is  preferable  and  generally  no  more  expensive.  It 
will  be  noted  that  the  maximum  bending  stresses  in  the  end  post  from  the  wind  forces  occur 
at  the  shoe  and  at  the  point  of  the  attachment  of  the  portal  strut,  and  also  that  the  portal 
strut  connections  must  be  designed  to  resist  this  maximum  moment. 

324.  The  Stress  Sheet. — The  sectional  areas  of  all  the  members  have  now  been  deter- 
mined and  the  stress  sheet  can  be  made.  This  sheet  should  show  the  dead  and  live  load 
stresses  on  each  member,  the  allowed  stresses  per  square  inch,  the  make-up  of  the  members, 
the  material  of  which  they  are  made,  whether  wrought-iron  or  steel,  the  live  and  dead  loads 
used,  the  specifications  under  which  the  design  was  made,  all  the  general  dimensions — such  as 
length  centre  to  centre  of  end  pins,  number  of  panels  and  length  of  each,  depth  centre  to 
centre  of  chords,  width  centre  to  centre  of  trusses  and  stringers,  the  skew,  if  any,  and  the 
alignment  of  the  track  over  the  bridge.  The  stress  sheet  for  the  150-foot  span  is  shown  in 
Fig.  361. 

325.  Details  at  the  Joints. — The  determination  of  the  sizes  of  pins  to  use  and  the 

arrangement  of  the  packing  on  a  pin  at  a  joint  is  most  conveniently  made  at  the  same  time 
the  sizes  of  the  pin  plates  or  bearings  are  proportioned.  Each  joint  will  be  designed  com- 
pletely as  we  proceed,  in  order  to  avoid  as  much  repetition  as  possible. 

326.  Joint  a. — The  pieces  meeting  at  this  joint  are  the  inclined  end  post,  aB  \  the  bottom 
chord,  ab  ;  the  shoe  ;  the  lateral  strut,  aa  ;  and  the  lateral  diagonal,  ab' .  The  members  of  the 
lateral  system,  na'  and  ab' ,  will  be  attached  to  the  end  post  as  close  to  the  pin  as  possible,  the 
aim  being  to  produce  the  least  bending  stress  from  an  eccentric  connection. 

A  preliminary  assumption  of  the  size  of  the  pin  must  first  be  made  and  the  thickness  of  bearings  re- 
quired for  this  size  determined.  The  bending  moments  produced  by  the  stresses  in  the  members  connecting 
on  the  pin  may  then  be  calculated,  and  if  the  pin  assumed  is  large  enougli  no  further  computation  is  neces- 
sary. There  are  cases  where  it  may,  however,  be  desirable  to  reduce  the  size  of  the  pin  in  order  to  use 
smaller  heads  on  the  eyebars,  but  it  is  assumed  here  that  the  desirable  and  smaller  size  of  pin  is  assumed 
originally-  The  maximum  diameter  of  the  head  for  a  steel  eyebar  should  not  be  over  2f  times  the  width 
of  the  bar  on  account  of  the  difficulty  in  upsetting  the  material.  The  desirable  diameter  of  head  is  about 
2\  times  the  width  of  bar,  and  as  an  excess  of  net  area  in  the  head  over  the  bar  of  from  30  to  40  per  cent 
is  required,  the  diameter  of  the  desirable  size  of  pin  is  about  nine  tenths  the  width  of  bar.  By  referring 
to  the  table  of  standard  steel  eyebars,  Chapter  XVII,  it  will  l)e  noted  that  two  sizes  of  heads  are 
generally  used  for  each  width  of  bar.  The  smaller  head  is  the  more  desirable  one  to  use  because  of 
its  cheapness  in  manufacture  and  as  it  requires  less  material.  The  thickness  of  the  eye  should  be  the 
same  as  the  body  of  the  bar  in  steel  eyebars.  The  smallest  pin  should  have  a  diameter  equal  to  three 
quarteis  of  the  width  of  the  bar  in  order  that  the  bearing  pressure  of  the  bar  on  the  pin  should  not  be  toe 
great.  It  is  custoniary  to  assume  the  size  of  the  pin  in  the  preliminary  calculations  from  the  above  con- 
siderations, as  the  cost  of  manufacture  is  affected  more  by  a  change  in  the  size  of  pin  than  in  the  amount  of 
material  required. 

The  allowed  extreme  fibre  stress  on  steel  pins  is,  by  the  specifications  under  which  this 
bridge  is  being  designed,  20  per  cent  more  than  that  specified  for  iron  pins,  or  18,000  lbs. 
per  square  inch.    The  pin  at  joint  a  will  be  assumed  to  be  4-}|  inches  in  diameter.  Tiie 


THE  COMPLETE  DESIGN  OF  A  SINGLE  TRACK  RAILWAY  BRIDGE. 


331 


332 


MODERN  FRAMED  STRUCTURES. 


allowed  bearing  pressure  per  square  inch  of  aB  on  the  pin  is  if  X  8300  =  12,450  lbs. 
The  stress  in  the  end  post  is  231,000  lbs.,  requiring    18^^^  square  inches  of   bearing  on 

the  pin.    The  thickness  of  the  bearing  required  —  i8j\  —  3I 

inches.  As  the  web  plates  bear  on  the  pin,  the  amount  of  addi- 
tional thickness  to  be  provided  is  3|  —  f  =  2-|  inches.  Plates  of 
the  following  thickness  will  be  used  ;  two  inch  and  two  -^-^  inch 
thick  on  the  outside  of  the  web  plates,  and  two  ^-^  inch  thick  on 
the  inside  of  the  webs.  They  will  be  arranged  as  shown  in  Fig. 
362.  Plates  b  are  made  the  same  thickness  as  the  top  angles. 
3^^'  Plates  a  are  as  thick  as  they  could  be  made  and  leave  room  enough 

to  drive  the  rivets  through  the  top  plate  and  angles.  Plates  c  have  to  make  up  the  re- 
quired thickness  of  bearing,  or  what  is  needed  beyond  that  used  in  the  web  plates,  a  and  b. 
Plates  c  will  be  the  hinge  or  lap  plates  (i.e.,  the  ones  which  extend  beyond  the  pin  and 
have  a  full  pin-hole  in  them). 

327.  The  Pin  Bearing  on  the  Shoe. — The  pressure  from  the  dead  and  live  loads  on 
the  shoe  is  vertical  and  is  equal  to  the  vertical  component  of  the  stress  in  the  end  post,  aB, 

28 

or  231,000  X  — —  =  172,500  lbs.    The  bearing  area  required  on  the  pin  is  172,500  12,450 

=  13.85  square  inches.  For  a  pin  4|f  inches  in  diameter  a  thickness  of  13.85  4.94  =  2.80 
inches  is  necessary.  A  hinge  plate  f  inch  thick  will  be  used  as  the  outside  plate  on  each 
rib  of  the  shoe.  The  clearance  between  the  outside  of  the  pin  plates  on  aB  and  the  inside  of 
these  hinge  plates  on  the  shoe  must  be  \  inch.  The  distance  between  the  inside  faces  of 
the  hinge  plates  must  be  therefore  17I  inches.  The  distance  centre  to  centre  of  the  ribs  of 
the  shoe  is  then  17I  +  f  —  lyV  =  I7tV  inches.  It  will  be  noted  that  each  rib  of  the  shoe  is 
made  ly\  inches  thick  in  order  to  make  them  both  alike,  and  to  use  no  plate  the  thickness  of 
which  is  measured  in  thirty-seconds  of  an  inch. 

328.  The  Calculation  of  the  Maximum  Bending  Moment  on  the  Pin  a. — The  bearings 
of  all  the  riveted  members  which  connect  on  the  pin  '  a '  have  now  been  determined  for  the 
assumed  pin,  and  the  bending  moment  on  the  pin  must  now  be  cal- 
culated to  see  whether  this  pin  is  large  enough.  The  centres  of 
bearings  of  the  various  members  will  be  locatec^  as  shown  in 
Fig-  363.  For  the  bending  moment  in  the  horizontal  plane  the 
pin  is  considered  as  a  beam  between  the  centres  of  the  bearings 
of  aB  and  loaded  with  two  loads  at  the  centres  of  bearing  of  the 
eyebars  ab* 

The  horizontal  component  of  the  stress  in  aB  is  154,000 
lbs.,  and  the  stress  in  each  bar  ab  is  77,000  lbs.  The  maximum 
bending  moment  from  the  horizontal  forces  on  the  pin  is  there- 

j  j-8           J  J  6 

fore  77,000  X     ^  ^         —  144,400  in.-lbs. 

For  the  bending  moment  in  the  vertical  plane  the  pin  is 
considered  as  a  beam  between  the  centres  of  the  bearings  of  the  Fig.  363. 

shoe  and  loaded  at  the  centres  of  the  bearings  of  aB.  The  vertical  component  of  the  stress 
in  aB  is  172,500  lbs.,  and  the  vertical  pressure  on  each  bearing  is  86,250  lbs.    The  maximum 

bending  moment  from  the  vertical  forces  is  therefore  86,250  X         ^          =  72,800  in.-lbs. 

The  maximum  moment  on  the  pin  is  the  resultant  of  these  two  moments,  or     144400'  +  72,800' 


*  Throughout  this  discussion  the  members  pf  the  truss  will  be  desigt»,a(ed.  Ijy  the  letters  at  their  extreme  ends,  as. 
shovvn  in  Fig.  355,   - 


THE  COMPLETE  DESIGN  OF  A  SINGLE  TRACK  RAILWAY  BRIDGE.  333 


=  162,000  in. -lbs.  This  requires  a  steel  pin  df-^^  inches  in  diameter,  allowing  18,000  lbs.  per 
square  inch  extreme  fibre  stress.    The  pin  \\\  inches  in  diameter  will  be  used. 

329.  Joint  JS. — Assume  the  pin  to  be  4^ f  inches  in  diameter,  the  same  as  used  at  a. 
The  thickness  of  bearing  required  for  aB  is  3f  inches  and  for  BC  is  238, 100 12,600== 

18.89-^-4.94=382  inches.  The  hinge  plates  on  aB  \^'\\\  be 
put  inside  and  those  on  BC  outside,  and  in  each  case  will  be 
I  inch  thick.  A  clearance  of  one  quarter  of  an  inch  will  be 
left  between  any  hinge  plate  and  the  nearest  member.  The 
packing  of  this  pin  and  the  distances  centre  to  centre  of  the 
bearings  of  the  several  members  which  attach  thereto  are  shown 
in  Fig.  364.  The  maximum  moment  on  this  pin  may  occur  at 
either  of  two  times:  ist,  when  the  stress  in  the  diagonal  Be  is  a 
maximum,  or,  2d,  when  the  stress  in  aB  is  a  maximum. 

1st.  WJien  the  stress  in  Be  is  a  maxifniim. — The  stress  on  Be 
is  then  149,400  and  the  stress  on  Bb  is  20,800  lbs.    From  the 
polygon  of  forces  at  this  point.  Fig.  365,  we  get  the  corresponding 
stresses  in  aB,  BC.    The  vertical  forces  arc  Bb  =  20,800,  Be  = 
Fig.  364.  1 11,600,  and  aB  =  132,400  lbs.    The  maximum  vertical  moment 

is  therefore  10,400  X  3tV  +  5 5, 800  X  i|  =  136,500  in. -lbs.*  The  horizontal  forces  are  Be  — 
99,600,  aB  =  118,200,  and  BC  =  217,800  lbs.  The  maximum  horizontal  moment  is  49,800  X 
2j\  -\-  59,100  X  if  =  152,800  in.-lbs.  The  resultant  mo- 
ment is  "V^  136,500' -(-  1 52,800"  =  205,000  in.-lbs.,  requiring  B6 
a  steel  pin  4i|  inches  in  diameter. 

2d.  JV/ien  tlie  stress  in  aB  is  a  maximum.  The  stress 
on  Bb  is  72,200  and  on  aB  =  222^400  lbs.*  From  the 
polygon  of  forces  acting  at  this  joint  we  get  Be  =  125,800 
and /^C  =  232,100  lbs.  The  vertical  forces  are  rt^' =  166,100, 
Bb  =  72,200,  and  Be  =  93,900  lbs.  The  vertical  moment 
is  36,100  X  3xV  +  46.950  X  i|  =  198,600  in.-lbs.  The 
horizontal  forces  are       =  148,300,  i?r  =  83,800,  and  232,100  lbs.    The  horizontal 

moment   is  41,900  X  2y\  +  74,150  X  if  =  141,600   in.-lbs.    The  resultant   moment  is 

V'  198,600'  -|-  141,600'  =  244,000  in.-lbs.,  requiring  a  steel  pin  5^'^  inches  in  diameter  which 
will  be  used. 

A  smaller  bending  moment  could  have  been  obtained  for  this  pin  by  putting  the  bars  Be  outside  of  aB 
and  BC,  but  it  is  alv/ays  preferable  to  pack  the  bars  inside  in  order  to  avoid  cutting  off  the  horizontal  legs 
of  the  bottom  angles  of  BC,  which  would  have  to  be  done  to  get  the  bars  close  to  the  bearings  of  aB  and  BC. 

The  thickness  of  the  bearings  required  for  aB  and  BC  will  be  a  little  less  than  what  was 
needed  for  the  4||-inch  pin  assumed,  but  owing  to  the  requirements  for  clearances  in  the  fit  of 
aB  and  BC  together  on  the  pin  the  thickness  used  will  be  maintained. 

330.  Joint  C. — A  pin  4y\  inches  in  diameter  will  be  assumed  in  order  to  use  small  heads 
on  the  5-inch  eyebars.  The  bearing  for  this  pin  on  CD  must  be  thick  enough  to  stand  the 
pressure  caused  by  the  horizontal  component  of  the  maximum  stress  in  Cel.  The  stress  in  BC 
is  transferred  to  CD  directly  by  a  butt  joint.  The  allowed  bearing  pressure  per  square  inch 
is  8400  X  =  12,600  lbs.  The  bearing  area  is  52^00 12,600  =  4^  square  inches,  or  for 
a  4-j-'^-inch  pin  the  thickness  required  is  4|-  4^"^^  =  W  inches.  The  web  plates  of  CD  are 
each  y\  inch  thick,  and  the  splice  plates  which  rivet  to  BC  and  CD  to  hold  them  in  position 
at  the  butt  joint  are  each  f  inch  thick.    There  is  necessarily  more  bearing  at  this  point  than 


Fig.  365. 


*  The  dead  load  concentration  at  this  joint  is  neglected  in  these  computations. 


334 


MODERN  FRAMED  STRUCTURES. 


is  required.  The  bearing  required  for  Cc  is  64,400  12,100  =  5|  square  inches.  The  thick- 
ness  required  is  5|  4^'^  =  inches.  One  plate  f  inch  thick  will  be  used  on  each  side  of 
the  post,  making  the  centre  to  centre  of  the  post  bearings  lof  inches.  The  usual  packing 
of  a  top  chord  pin  is  as  shown  in  Fig.  366,  which  is  a  sketch  of  the  joint  C.  The  maximum 
moment  on  the  pin  occurs  when  the  diagonal  Cd  has  its  maximum  stress.  The  forces  acting 
on  the  pin  then  are  Cd  —  77,800,  Cc  =  64,400,  CD  —  51,900  horizontal  ■and  CD  =  6400  vertical 
(assuming  the  dead  load  panel  concentration  to  be  applied  at  the  bearings  of  CD).  The 
vertical  forces  are  Cd  =  58,000,  Cc  =  64,400,  CD  =  6400,  and  the  vertical  bending  moment  is 
3200  X  2^1  +  29,000  X  tI^  =  34,900  in.-lbs.  The  horizontal  forces  are  Cd  =  51,900  and 
CD  =  c,  I, goo,  and  the  horizontal  bending  moment  is  25,950  X  i|f  =  38,100  in.-lbs.  The 

resultant  bending  moment  is  V 34,900""  +  38,100'  =  51,700  in.-lbs.  This  would  allow  the  use 
of  a  3j-V''ich  pin,  but  a  pin  3-^!^  inches  in  diameter  will  be  used,  as  this  is  the  smallest  pin 
allowable  for  a  5-inch  bar.  The  thickness  of  the  bearing  plates  for  Cc  will  be  5|^  —  3^1  =  1 3 
inches.    One  plate  \^  inch  thick  will  be  used  on  each  side  of  the  post. 


Fig.  366.  Fig.  367. 


331.  Joint  -D. — A  pin  3i|  inches  in  diameter  will  be  used  and  the  pin  packed  as  shown 
in  Fig.  367.  The  counter-ties  are  put  on  the  middle  of  the  pin,  the  webs  of  the  channels  in 
Dd  being  cut  to  let  them  pass.  As  there  is  only  one  rod  each  way,  the  bars  are  necessarily 
packed  off  of  the  centre,  but  the  stresses  in  them  are  so  small  that  the  effect  is  insignificant. 
The  bearing  on  the  pin  required  for  Dd  is  19,500-=-  12,900=  1. 51  square  inches  or  y\  inch 
thick.    One  plate  f  inch  thick  will  be  used  on  each  side  of  the  post. 

332.  Joint  d. — A  pin  5^  inches  in  diameter  will  be  assumed.  The  only  member  for 
which  the  bearings  must  be  determined  is  Dd.  The  maximum  stress  in  the  post  at  the 
bottom  is  78,600  lbs.,  and  occurs  when  the  floor-beam  has  its  maximum  concentration 
from  the  live  load.  The  dead  load  panel  concentration  is  also  assumed  as  acting  at  the 
centres  of  bearings  of  Dd.    The  allowed  bearing  stress  per  square  inch  on  the  pin  at  the 

foot  of  the  post  is  li  X  6500(1  4-  ^^'^OQ  j  _  jj  jqo  lbs.     The  bearing  area  required  is 

^       78,600 ' 

78,600 -7-  11,100  =  7,08  square  inches,  or  7,08 =  if  inches  thick.  Two  plates  each  | 
inch  thick  will  be  used  on  each  side  of  the  post.  The  packing  of  the  several  members  on 
the  pin  will  be  as  shown  in  Fig.  368. 

The  maximum  bending  moment  occurs  when  cd  has  its  maximum  stress.  The  stresses 
then  are  cd  =  238,100,  c'd'  =  234,500,  Cd  =  47,200,  Cd'*  =  52,700,  and  Dd  =  74,500  lbs. 


*  Here  the  primed  letters  represent  corresponding  joints  on  the  right  of  the  centre  of  the  bridge. 


THE  COMPLETE  DESIGN  OE  A  SINGLE  TRACK  RAILWAY  BRIDGE.  335 


The  vertical  stresses  are  Cd  =  35,200,  C'd'  —  39,300,  and  Dd  —  74,500  lbs.    The  hori- 
zontal stresses  are  cd  =  238,100,  c'd'  —  234,500,  Cd  =  31,500,  and 
C'd'  =  35,100  lbs.    The  vertical  moments  are  as  follows : 

About  Cd  =  19,650  X  yf  =  18,400  inch-pounds. 

"     Dd  —  19,650  X  2y\  +  17,600  X  li  =  74,300  inch-pounds. 

The  horizontal  moments  are  as  follows : 


About  c'd'  =  119,050  X 
«    Cd'  =  1 19,050  X  3t\ 
pounds. 

"      Cd=  1 19,050  X  48  —  1 17.250  X  2j\ 
203,500  inch-pounds. 


21  5,800  inch-pounds ; 
117,250  X  if  =  218,300  inch- 


17.550  X  If 


The  resultant  bending  moment  at  Dd  is  V  74,300'  +  203,500'= 
217,000.  The  maximum  bending  moment  is  at  C'd'  and  is  due 
to  the  chord  stresses  alone.  The  diameter  of  the  smallest  pin 
which  could  be  used  without  exceeding  the  allowed  extreme  fibre 
stress  is  5ji^  inches.  5y3^  inches  will  be  the  diameter  of  the  pin 
used. 

333.  Joint  c. — In  order  to  keep  the  sizes  of  the  pins  as 
nearly  alike  as  possible  a  pin  5^^-   inches  in  diameter  will  be 


Fig.  368. 


li  X  65001  I  + 


The  maximum  stress  in  the  post  l>elow  the  floor-beam 


I 


I 


■4 


1 


assumed.    The  thickness  of  bearing  for  Cc  must  be  determined. 

The  allowed  bearing  pressure  per  square  inch  on  the  pin  for  the  foot  of  the  post  is 

28,8oo\ 

 =  12,200  lbs. 

1 1 1,500/ 

is  equal  to  the  vertical  component  of  the  maximum  stress  in  Be.    The  minimum  stress  is  the 

vertical  component  of  the  dcdd  load  stress  in  Be,  assuming  as  is  cus- 
tomary that  the  dead  load  panel  concentration  is  applied  to  the  pin 
through  the  post.  The  bearing  area  required  at  the  foot  of  the 
post  is  1 11,500 -^-  12,200  =  9.14  square  inches.  The  thickness  of 
bearing  is  9.14 -f-  Sts  —  'tI  inches.  One  plate  f  inch  thick  and 
one  plate  inch  thick  will  be  used  on  each  side  of  the  post. 
The  packing  of  the  members  on  the  joint  will  be  as  shown  in  Fig. 
369- 

There  are  two  cases  to  be  examined  for  maximum  bending 
moment :  1st,  when  Be  is  a  maximum,  and  2d,  when  i>c  is  a  maxi- 
mum. 

1st.  W/icfi  Be  is  a  maximum. — The  stresses  then  existing  arc 
Be  =  149,400,  Cc  =  111,500,  i>e  =  123,800,  and  cd  =  223,400  lbs. 
The  vertical  forces  are  Be  =  111,500  and  Cc  —  111,500.  The  ver- 
tical bending  moment  is  55,750  X  i||  =  102,800  in.-lbs.  The 
horizontal  forces  are  Be  =•■  99,600,  dc  - 
lbs.    The  horizontal  moments  are: 


Fig.  369. 


About  ed 
"  Be 


123,800,  and  cd  =  223,400 

61,900  X  if  =  100,600  in.-lbs., 

61,900  X  3i  —  111,700  X  i|  =  19,700  in.-lbs. 


The  resultant  bending  moment  at  any  point  of  the  pin  between  the  bearings  of  Cc  is 
y/ 102,800"  -{-  19,700"  =  104,700  in.-lbs. 

2d.  Wken  be  is  a  maxiumm. — The  stresses  then  existing  are  Be  =  120,300,  dc  =  153,900, 


Modern  framed  structures. 


cd  =  234,100,  and  Cc  =  89,800  lbs.  The  vertical  lorces  are  Be  =  89,800  and  Cc  =  89,800  lbs. 
The  vertical  bending  moment  is  44,900  X  if  J  =  82,800  in.-lbs.  The  horizontal  forces  are  : 
£c  =80,200,  6c  =  153,900,  and  cd  =  234,100.    The  horizontal  bending  moments  are  • 

About  cd  =  76,950  X  if  =  125,000  in.  lbs. 

"    Be  =  76,950  X  3i  —  117,050  X  if  =  59,900  inch-lbs. 

The  resulting  bending  moment  at  any  section  of  the  pin  between  the  bearings  of  Cc  is 

2,800  -f-  59,900  =  102,200  in.-lbs.  A  pin  4^'^  inches  in  diameter,  the  smallest  allowable 
for  a  six-inch  eyebar,  will  be  used  at  this  joint.  The  thickness  of  the  pin  bearing  for  Cc  will 
be  increased  to  9. 14 4.44  =  2.06  inches.  One  plate  f  inch  thick  and  one  plate  ^  inch 
thick  will  be  used  on  each  side  of  the  post. 

334.  Joint  6, — In  order  to  understand  the  packing  of  this  joint  and  the  critical  stresses, 

it  is  necessary  to  completely  design  the  details  of  the 
hanger  Bd  to  which  the  floor-beam  and  lateral  rods  are 
attached.  Fig.  370  is  the  detail  sketch  of  this  joint.  The 
floor-beam  is  riveted  to  a  plate,  A.  This  plate  A  is  held 
up  by  the  bars  Bb.  The  lateral  rods  are  attached  to  the 
lower  flange  of  the  beam.  The  permissible  tensile  stress 
per  square  inch  on  the  floor-beam  hanger,  as  plate  A  is 
called,  is  5000  lbs.  per  square  inch.  The  stresses  in  the 
plate  around  the  pin  are  similar  to  those  in  an  eyebar 
head.  The  net  area  of  the  plate  A  at  a  horizontal  section, 
BC,  through  pin  6^  must  be  at  least  25  per  cent  in  excess  of 
the  net  area  required  at  5000  lbs.  per  square  inch.  The 
net  area  of  the  vertical  section  above  the  pin  and  on  the 
line  joining  the  pin  centres  of  and  B  must  be  three 
quarters  of  the  net  area  of  the  horizontal  section  BC,  as 
the  tensile  stress  on  this  area  acts  perpendicular  to  the 
fibres  of  the  iron,  and  wrought-iron  is  about  two  thirds  as 
strong  in  tension  across  the  grain  or  fibre  as  it  is  with  the 
grain.    The  bearing  of  the  pin      on  the  hanger  A  must 

/     I  i2,8oo\ 

not  exceed  the  specified  stress  of  6500  1 1  +   if  = 

^  ^  72,200/ 

11,500  lbs.  per  square  inch. 

The  hanger  is  subjected  to  a  bending  stress  from  the 

chord  component  of  the  lateral  rods  ad',  be'  *  This  stress 

is  transferred  to  the  pins  b^  and  b  by  the  plate  A.  The 

part  transferred  to  the  pin     is  taken  up  by  the  rods  ab^  or 

T    b^c  and  Bb.    The  stress  from  this  cause  in  Bb  is  small  and 

always  neglected  in  determining  the  area  of  Bb.  The 

areas  of  the  rods  ab^ ,  b,c  are  determined  by  this  stress 

alone.    The  part  transferred  to  the  pin  b  is  taken  up  by 

the  chord  bars  be. 

37°-  The  pin  b,  is  3f|  inches  in  diameter.    The  bearing 

required  on  this  pin  is  72,200 -^  11,500  =  6.29  square  inches,  or  1.6  inches  thick.  The  net 
area  of  the  hanger  required  at  a  horizontal  section  at  the  top  of  the  beam  is  72,200^  5000  = 
14.44  square  inches,  which  is  satisfied  by  using  a  plate  16  X  ItV  inches,  making  allowance 
for  two  1-inch  rivet-holes.  The  net  area  at  the  section  BC  must  be  14.44  X  1.25  =  18.05 
square  inches,  requiring  a  plate  16  X  li  inches,  making  allowance  for  the  reduction  of 

*  Here  the  primed  leuers  represent  corresponding  joints  in  the  other  truss. 


T 


THE  COMPLETE  DESIGN  OF  A  SIMgLR  TRACK  RAILWAY  BRIDGE.  337 


the  available  area  by  the  pin-hole.  The  bearing  requires  1.6  inches  thickness.  The  plate  A 
will  be  built  of  two  plates  16  X  2  16  X  inches  each.  The  required  bearing  on  pin 
and  the  area  of  the  section  BC  will  be  satisfied  by  the  addition  of  two  plates,  d,  f 
inch  thick,  one  on  each  side  of  A.  The  area  of  the  vertical  section  above  the  pin 
must  be  f  of  18.05  =  13-54  square  inches.  The  total  thickness  of  plates  A  and  d  now  is 
\-Y^-\-\  =  lyf.  The  distance  from  the  top  side  of  the  pin-hole  b^  to  the  end  on  the  hanger 
must  be,  in  order  to  give  the  required  area  of  13.54  square  inches,  13.54  -J-  1.8 1  =  7.5  inches. 

The  chord  component  of  the  stress  in  the  lateral  rod  ab'  is  61,400  X  — >^  =  51,700  lbs., 

2Q.Oo 


•        ,  J      /         r  II  25.75 

causmg  a  stress  m  the  rods  a.c  01  51,700  X  —  X  

s  1       3  ./  ^  25.0 


7900  lbs.    One  rod  one  inch  square 
is  the  smallest  rod  allowed,  and  this  size  will  be  used  for  b,c  and  ab^.    The  bending  moment 

7QOO  X  I 

on  the  pin  b^c  will  be   I2L  =  2840  in. -lbs.    A  pin  ly^^  inches  in  diameter  will  be  used. 


The  plates  d  are  extended  to  take  these  pins,  making  the  distance  centre  to  centre  of  bearings 
for  the  pins       -J-  f  =  ly'^,  as  used  in  the  above  computation. 

5 1  700  X  6  ?  X  II 

The  maximum  bending  moment  on  the  plate  A  from  the  lateral  rods  is  ^—^  

=  484,200  in.-lbs.  The  permissible  fibre  stress  per  square  inch  will  be  taken  as  12,000  lbs., 
the  maximum  stress  allowed  on  short  columns  for  the  lateral  system.  The  bending  moment 
requires  a  plate  16  inches  wide  and  |f  inch  thick.  Plate  A  will  be  made  i-^^  inches  thick 
throughout. 

Pin  b  may  now  be  deterrtiined.  As  there  are  no  members  on  this  pin  for  which  the 
bearings  must  be  proportioned,  the  bending  moment  may  be  computed  and  the  required  pin 
selected  at  once.    The  packing  of  this  pin  is  shown  in  Fig.  370. 

The  bars  a6  and  dc  are  put  so  far  from  the  hanger  A  because  it  is  desirable  to  have  the  bars  coincide  as 
nearly  as  possible  with  straight  lines  drawn  from  the  centre 
of  each  bar  on  the  pin  a  to  the  centre  of  each  bar  on  pin  c. 
This  will  cause  the  least  deviation  of  the  bars  from  a  line 
parallel  to  the  centre  line  of  the  bottom  chord.  It  is  a 
common  requirement  of  good  construction  that  the  bars 
run  as  nearly  parallel  to  the  centre  line  of  the  chord  as  prac- 
ticable. If  the  bars  deviate  from  a  line  parallel  to  the  centre 
line  of  the  chord  more  than  one  eighth  of  an  incli  to  the 
foot,  the  bars  must  be  bent  or  the  pin-holes  in  the  bars 
bored  as  shown  in  Fig.  371.  This  involves  additional  ex- 
pense and  is  not  desirable  construction.  Pj(, 

The  vertical  moment  on  the  pin  due  to  the  weight  of  the  chord  bars  alone  is  small  and 
will  be  neglected.  The  maximum  horizontal  bending  moment  on  the  pin  from  the  dead  and 
live  load  stresses  in  the  chords  ab,  be  is  76,950  X  lyV  =  '  10,700  iii.-lbs.  This  would  require 
a  steel  pin  3i|  inches  in  diameter.    The  chord  component  from  the  wind  stress  in  the  lateral 

,    7/  ,      ,•  r  63  17.81 

rod  ab  causes  a  bending  moment  of  51,700  X  —  X  =  iq6,ooo  m.-lbs.,  making  a  total 

74  4 

bending  moment  of  i  10,700  +  196,000  =  306,700  in.-lbs.  from  the  dead,  live,  and  wind 
stresses.  For  this  combination  it  is  usual  to  increase  the  permissible  extreme  fibre  stress  pei 
square  inch  one  half,  or  in  the  present  case  to  27,000  lbs.    This  would  require  a  steel  pin  4J-I 


inches  in  diameter  which  will  be  used. 

The  bending  moment  on  the  pin  is 


72,200  X  3i 


56,400  in.-lbs. 


The  sizes  of  all  the  truss  pins  have  now  been  determined,  and  the  next  problem  is  to 


33^ 


MODERN  FRAMED  STRUCTURES. 


E 


o 

o  < 

o 

J  o 
o 

(  o 
0 

determine  the  lengths  and  number  of  rivets  required  for  the  bearing  plates  on  the  various 
members. 

335.  The  Lengths  of  Bearing  or  Pin  Plates  are  determined  by  the  following  con- 
siderations : 

1st.  Each  plate  must  be  long  enough  to  take  sufficient  rivets  to  transfer  the  stress  in  the 
plate  to  the  main  section. 

The  stress  in  a  pin  plate  may  be  the  accumulated  stresses  in  a  number  of  plates  outside  of  the  plate  in 
question,  but  which  must  pass  through  this  plate  to  reach  the  main  section.  Where  there  are  a  number  of 
plates  the  stress  per  square  inch  on  the  plates  nearer  the  main  section  increases  slightly,  hence  it  is  always 
better  to  use  the  thicker  plates  nearest  the  main  section. 

It  is  not  sufficient  to  merely  put  the  required  number  of  rivets  through  the  plate,  but  there  must  also  be 
enough  rivets  to  resist  any  bending  moment  due  to  an  eccentric  application  of  the  stress  with  respect  to  the 
centre  of  the  group  of  resisting  rivets.  In  order  to  fulfil  this  requirement  it  is  often  necessary  to  use  more 
rivets  than  the  number  determined  by  dividing  the  bearing  stress  on  the  plate  by  the  value  of  one  rivet. 

2d.  The  distance  from  the  edge  of  the  pin-hole  to  the  end  of  the  plate  must  alvi^ays  be 
great  enough  to  offer  sufficient  resistance  against  splitting  on  the  line 
AB,  Fig.  372,  the  dangerous  section  being  the  one  which  has  the  least  net 
area  or  is  the  most  cut  up  by  rivet-holes. 

The  best  way,  probably,  to  determine  the  distance  C,  Fig.  372,  is  to  consider  the 
plate  as  a  beam  balanced  over  the  centre  of  the  pin  and  acted  upon  by  forces  each 
equal  to  the  value  of  one  rivet  applied  at  the  centres  of  the  rivets. 

It  is  a  safe  rule  to  make  the  width  of  pin  plate  not  less  than  three  fourths  of  the 
Fig.  372.         length.    This  question  is  often  overlooked  in  designing,  but  even  as  a  question  of 
appearance  only  it  deserves  attention. 

3d.  At  least  one  pin  plate  on  each  web  plate  must  be  used  to  reinforce  the  web  plate 
until  the  stresses  are  distributed  over  the  entire  area  of  the  member.  Thus,  in  aB,  the  entire 
stress  is  first  received  from  the  pin  on  the  pin  plates  and  web.  The  top  plate  and  angles 
must  receive  their  proportion  of  the  stress  through  the  rivets  connecting  them  to  the  web 
plates,  and  until  this  stress  is  transferred  the  web  plate  alone  would  not  be  sufficient  to  take  it 
without  being  subjected  to  a  greater  stress  than  is  allowed.  The  amount  of  stress  taken  by 
the  top  plates  and  angles  of  aB  is  such  a  proportion  of  the  total  stress  on  aB  as  the  area  of 
the  top  plate  and  angles  is  of  the  total  area,  or  101,300  lbs.  To  transfer  this  amount 
requires  thirty-four  rivets  in  single  shear,  the  diameter  of  the  rivet  being  |  inch.  It  will 
readily  be  seen  that  it  is  advantageous  and  economical  to  use  as  many  rivets  in  double 
shear  as  possible  in  order  to  transfer  this  stress  in  the  shortest  distance.  For  this  reason 
plate  a,  Fig.  373,  was  made  wide  enough  to  take  the  rivets  through  the  top  angles.  The 
shortest  rivet  spaces  allowable  should  always  be  used  at  the  ends  of  compression  members  and 
continued  until  the  stress  has  had  an  apportunity  to  distribute  itself  over  the  entire  area  of 
the  member.    Plates  b,  Fig.  373,  are  made  long  in  order  to  reinforce  the  web  plates. 

336.  Pin  Plates  on  aB  at  Joint  — The  pin  plates  are  a,  inch  thick,  b,  inch  thick, 
and  c,  y\  inch  thick  (see  Fig.  373).  The  number  of  rivets  through  these  plates  attaching  them 
to  the  main  section  of  the  end  post  is  as  follows: 

231,000  X  A  «  ,  , 

Plate  a  1 — —  =  19,250     3000  =  7  |-mch  rivets  m  smgle  shear. 

231,000  X  A 

«      l>   i5       34^650 -r-  3000  =  12     «  «       «     "  ** 


231,000  X  A 

.  =  34,650  ^  3000  =  12 


«       <(  It 


THE  COMPLETE  DESIGN  OF  A  SINGLE  TRACK  RAILWAY  BRIDGE.  339 


This  number  of  rivets  will  be  sufficient  if  the  centre  of  the  group  of  resisting  rivets  is  on 
the  centre  line  connecting  the  pins  a  and  B.    In  addition  to  the  above  it  must  be  observed 

ViirzrrBV  _  53^^  ^  ^OOO  =  18  rivets  in  single  shear  attaching 


that  there  must  be 


3i 


a  and  b  to  the  main  section,  and  finally  there  must  be  enough  rivets  attaching  a,  b,  and  c  to 
the  main  section  to  transfer  the  total  bearing  pressure  on  the  three  plates  to  the  main  section 
without  exceeding  the  allowed  bearing  pressure  of  rivets. 


_  r>  -  ^  ^  0  0  ^  Q  n-  n.-  n — o — q — e 

1 — 0 — Q — n  n  , 

>-v   0  0 

oj    0     0     0  0 

v_y  0  0 

•0000 

\o  0  0 

oi    0     0     0  0 

6  0 

0000  0 

V  u  u  u  u  u' 

- — 


FiG.  373. 


In  plate  a  thirteen  rivets  are  used.  This  is  more  than  would  be  necessary  for  this  plate, 
but  if  the  plate  were  shortened  3  inches  both  the  first  and  second  conditions  mentioned  as 
determining  the  length  of  pin  plates  would  not  have  been  fulfilled.  Plate  b  is  made  long  in 
order  to  stiffen  the  web  in  accordance  with  the  requirements  of  the  third  condition.  Plate  c 
has  seventeen  rivets,  which  is  a  few  more  than  necessary  to  fulfil  the  first  condition,  but  if  it 
were  made  3  inches  shorter  and  onif»thirte||^^ets  used  the  rivets  would  be  overstressed. 
Plate  c  is  also  the  hinge  plate.  The  distance  from  the  edge  of  the  pin-hole  to  the  end  of 
the  plate  should  be  not  less  than  2  inches. 

337.  Pin  Plates  on  aB  at  Joint  B. — The  plates  are  a,  inch  thick,  b,  \  inch  thick, 
and  c,  f  inch  thick.  Plate  c  is  the  hinge  plate.  Assuming  the  centre  of  the  group  of  resisting 
rivets  to  be  on  the  centre  line  joining  the  pins  a  and  B,  the  number  of  rivets  required  to  attach 
these  plates  to  the  main  section  singly  is  as  follows : 


Plate  a  . 


"  b. 


231,000  X  TB 

3l 

231,000  X  i 


=  34,650  3000  —  12  rivets  in  single  shear. 
=  30,800  -r-  3000  =11"     "     "  " 


231,000  X  I 


=  23,100 3000  r=     8    "  " 


F'g-  374  is  a  detail  sketch  of  the  bearing  of  aB  on  pin  B.  Plate  a  is  made  long  enough 
to  fulfil  the  third  condition.    Plate  b  has  twenty-two  rivets  attaching  it  to  the  web  plate.  As 


< 

a- 
< 


1  iCi  r\  —         iJi  ^li  ^     r-\   £2i — a. — r\  (X—C^  Q  C\  _ 

r 

0 

0 

0      0      0      0|  0  0;o  0  /^~\ 

0 

D 

0   ,0    0  j  0  Oi  0  0  \~y 

0 

0 

0     0     0     oio  oloo  /^.^ 

P 

0  0 

0 

P,o^Q^o^o^o„q  0  o;o  07 

'  u  u  ^  u     '  u  ' 

Fig.  374. 


b  must  have  enough  rivets  to  transfer  the  bearing  pressure  on  both  b  and  c.  it  will  be  seen  that 
it  is  as  short  as  it  could  have  been  made.  Plate  c  is  the  hinge  plate,  and  has  ten  rivets  when 
but  eight  are  necessary. 

338.  Pin  Plates  on  BC  at  Joint  B.— The  plates  are  a,  f  inch  thick,  b,  f  inch  thick, 


340 


MODERN  PR  A  MED  STRUCTURES. 


c,  y\  inch  thick,  and  d,  f  inch  thick.  Plate  a  is  the  hinge  plate.  The  number  of  rivets  required 
to  transfer  the  stresses  on  these  plates  to  the  main  section  is  as  follows: 


Plate  a. 


Plates 


Plates  a,  b,  and  c . . . . 
Plate  d  


238,100  X  I 
238,100  X  I 

08 

238,100  X  It\ 
238,100  X  I 


23,040     3000  =    8  |-inch  rivets  in  single  shear. 


are  met. 


^  r^..A  rs       r\  r\  c\  n  ri-ri_r> — r>— r>— r>  Cs  Q  , 

0  6  0T6 
vJ   0  0  ojo 
0  0  0:0 

00     0  0 
0  0 

00     0  0 

\0  0  0;0, 

000000000000  0 

=  16 

<<  (i 

«    «  « 

=  25 

_  Q 

<<  « 
((  <i 

(i    «  « 

lengths  of  pin 

plates  for  this 

( 

1 ' 

>  < 

< — a — ^ 

[j  E^a 

)  < 
)  < 
i  *  *■ 

|L6 

Fig.  BB-  • 

339-  Intermediate  Top  Chord  Joint. — The  top  chord  sections  are  spliced  as  shown  in 
Plate  I.  The  splice  plates  on  the  web  usually  take  one  or  two  rows  of  rivets  beyond  the 
splice.  The  splice  plate  on  the  top  plate  usually  has  but  one  row  of  rivets.  The  duty  of  the 
splice  is  to  hold  the  spliced  sections  truly  in  line,  and  is  not  relied  on  to  transfer  any  of  the 
stress.  The  ends  of  the  joined  sections  are  planed  oH  in  order  to  secure  a  perfect  bearing 
throughout  the  joint  ;  the  transfer  of  the  stress  is  supposed  to  be  done  by  the  abutting 
ends.  The  position  of  the  splice  is  determined  as  follows  :  The  splice  must  be  as  near  the  pin 
as  possible,  and  also  there  must  be  clearance  enough  to  allow  the  field  rivets  through  the  web 
plates  to  be  driven  easily.  The  splice  is  usually  located  by  the  clearance  lines  of  the  post  or 
tie-bars,  as  they  are  the  only  members  which  interfere  with  the  driving  of  the  field  rivets.  It 
is  customary  now  to  make  the  splice  on  the  side  of  the  joint  nearer  the  end  of  the  truss,  but 
this  is  dependent  upon  the  method  of  erection. 

340.  Post  Cc. — At  the  top  pin,  C,  the  bearing  of  this  post  is  as  previously  given,  l  ^  inches 
thick,  and  is  made  up  of  two  plates  f  inch  thick,  one  on  each  side  of  the  post.  The  stress 
on  each  plate  is  32,200  lbs.,  requiring  eleven  |-inch  rivets  in  single  shear  to  transfer  this  stress 
to  the  channels.  Twelve  rivets  are  used  in  each  plate,  or  six  through  each  plate  and  flange  of 
the  channel. 

At  the  bottom  pin,  c,  the  required  bearing  is  2.06  inches  thick,  and  one  plate  ^  inch 
and  one  plate  inch  thick  are  used  on  each  side  of  the  post.  On  the  side  of  the  post 
nearer  the  track  the  f-inch  plate  is  extended  in  order  to  attach  the  floor-beam  to  it.  The 

111,500  X  ^ 


number  of  rivets  required  to  attach  the  -f^-inch  plate  is 


2i 


=  36,100  -T-  3000  =  12. 


The  number  required  to  attach  the  two  plates  on  one  side  of  the  post  to  the  channels  is 
55,750  3000  =  19.  Twenty  rivets  will  be  used  through  the  pin  plates  and  channel  flanges  on 
the  outside  of  the  post,  and  on  the  inside,  or  track  side,  the  rivets  will  be  spaced  to  match 
the  floor-beam  rivets.    See  Plate  I. 

354.  Post  r>d. — At  pin  D  the  bearing  used  is  f  inch  thick,  or  one  |-inch  plate  on  each 
side  of  the  post.    The  number  of  rivets  required  to  attach  one  of  these  plates  to  the  flanges 


THE  COMPLETE  DESIGN  OF  A  SINGLE  TRACK  RAILWAY  BRIDGE.  341 


is  9750  -i-  3000  =  4.    Six  rivets  are  used  through  each  flange  of  the  channels  and  the  pin 
'    plates,  as  the  pitch  is  usually  kept  constant  at  3  inches  centre  to  centre  of  rivets  and  the 
plates  are  extended  so  as  to  take  at  least  two  rivets  beyond  the  point  where  the  web  of  the 
channels  is  cut  to  allow  the  counter-rods  to  pass. 

At  pin  d  the  bearing  is  \\  inches  thick,  made  up  of  four  plates  each  |  inch  thick,  two  on 
each  side  of  the  post.  One  of  the  plates  is  extended  upwards,  to  take  the  floor-beam  con- 
nection. The  number  of  rivets  required  to  attach  one  |-iiich  plate  is  19,650  3000  =  7,  and 
the  number  required  to  attach  the  two  plates  on  one  side  of  the  post  to  the  channel  flanges  is 
39,300  ^  3000  =  14. 

342.  The  End  Shoes  are  shown  in  Fig.  376.  The  thickness  of  pin  bearing  required  has 
been  found  to  be  2\  inches  thick.    The  bearing  pressure  on  the  masonry  is  limited  to  300  lbs. 


per  square  inch,  requiring  575  square  inches.  The  number  and  length  of  rollers  under  the 
shoe  at  the  expansion  end  is  determined  by  the  formula  /  =  250c/,  wlierc  /  is  the  allowed 
pressure  per  linear  inch  on  the  roller  and  d  is  the  diameter  of  the  roller.  For  d  —  p  z=.  750. 
Therefore  172,500  -=-  750  =  230  inches  of  3-inch  rollers  are  required. 

The  height  of  the  shoe  from  the  sole  plate  to  the  pin  must  be  enough  to  make  the  ribs  of 
the  shoe  stiff  enough  to  distribute  the  pressure  uniformly  over  the  masonry  or  the  rollers. 
The  sole  and  bed  plates  are  usually  made  f  inch  thick  and  large  enough  to  give  the  required 
bearing  area  on  the  masonry.  The  end  reactions  of  both  lateral  systems  must  be  transferred 
to  the  masonry  through  the  shoe.  This  requires  that  they  be  stiff  laterally.  This  stiffness  is 
usually  obtained  by  the  use  of  a  web  connecting  the  two  ribs  of  the  shoe,  or  by  a  plate  over 
the  ends  of  the  ribs  corresponding  to  the  top  plate  of  the  end  post. 

343.  Tie  Plates  and  Latticing. — The  specifications  require  that  the  tie  plate  shall  be 
long  enough  to  take  one  quarter  of  the  total  stress  on  the  member  through  the  rivets  in  one 

238.100 

segment.    This  requires  plates  on  aB  or  BC  long  enough  to  take  —  '-  3000  =  20  rivets 

or  5  feet  long.    The  thickness  of  the  tie  plates  is  also  limited  to  one  fortieth  of  the  distance 


342 


MODERN  FRAMED  STRUCTURES, 


between  the  rivets  in  the  two  segments  which  the  tie  plate  joins.  For  the  end  posts  and  top 
chords  this  requires  plates  \  inch  thick. 

The  usual  specification  for  tie  plates  requires  that  they  shall  have  a  length  equal  to  one  and  a  hall 
times  the  width  of  the  member.  For  aB  this  would  be  33  inches.  A  good  rule,  and  one  which  requires 
less  material,  is  to  make  the  tie  plates  on  the  inclined  end  posts,  vertical  posts,  and  on  the  top  chords,  at 
the  extreme  ends  only,  square,  and  those  at  the  ends  of  the  intermediate  top  chord  sections  12  inches  wide 
The  thickness  need  never  be  over  f  inch,  and  when  the  distance  between  the  rivets  connecting  the  plates  to 
the  two  segments  is  over  20  inches,  stiffener  angles  may  be  riveted  on  the  plates  and  a  saving  in  material 
made. 

The  pin  plates  on  the  vertical  posts  serve  the  purpose  of  tie  plates. 

The  width  of  lattice  bars  is  fixed  by  the  specifications  at  2^  inches  for  the  end  post  and 
top  chord,  2\  inches  for  post  Cc,  and  2  inches  for  post  Dd.  The  thickness  is  limited  to  one 
fortieth  of  the  distance  between  rivets  for  single  lattice,  and  one  sixtieth  of  this  distance  for 
double  lattice  riveted  at  the  intersection  of  the  bars.  Bars  2^^"  X  will  be  used  for  aB,  BC, 
and  CD,  2\"  X  |"  for  Cc  and  2"  X  A"  fo^"  The  tie  plates  for  the  top  struts  will  be 

made,  as  usual,  6  inches  wide  and  y^g-  inch  thick,  the  lattice  bars  2  inches  by  -^-^  inch. 

344.  Details  of  the  Floor-beams  and  Stringers  were  determined  when  the  calculations 
were  made  of  the  flange  sections.  The  number  of  rivets  required  in  the  connections  of  the 
stringers  to  the  beams  and  of  the  beams  to  the  posts  were  previously  computed.  The  detail 
sketches  will  sufficiently  explain  the  mode  of  construction.  The  end  stringers  must  have 
bearing  enough  on  the  masonry  to  keep  the  pressure  within  the  specified  limits.  They  must 
also  be  braced  together  by  an  X-frame.    This  detail  is  shown  completely  in  Plate  I. 

345.  The  Details  of  the  Lateral  System. — The  lateral  bracing  may  be  truly  said  to 
be  the  bete  noir  of  the  bridge  designer.  It  is  impossible  to  make  any  attachment  to  the 
trusses  which  will  cause  no  secondary  stresses.  The  method  of  attaching  the  top  lateral 
system  used  for  this  truss  is  one  of  the  best  in  use,  but  advantage  is  taken  of  the  fact  that  the 
stresses  are  small  and  the  members  of  the  truss  which  act  as  chords  for  the  wind  truss  are 
large  and  very  little  affected  by  an  eccentric  attachment.  The  details  of  the  lower  lateral 
system  are  very  common  practice  and  may  be  said  to  be  as  good  as  any  in  common  use.  The 
attachment  was  made  so  far  from  the  lower  chord  in  order  to  avoid  cutting  away  the  lower 
flange  angle  of  the  floor-beam  to  let  the  tie  bars  pass.  This  will  be  understood  by  referring 
to  the  detail  sketch  of  joint  c.  This  eccentricity  produces  a  bending  moment  in  aB,  Cc,  and 
Dd  practically  equal  to  the  chord  component  of  the  lateral  rod  into  the  distance  from  the 
plane  of  the  lateral  system  to  the  plane  of  the  lower  chord  pins.  A  careful  analysis  of  the 
resulting  stresses  will  show,  however,  that  the  stresses  per  square  inch  in  these  members  is 
within  the  allowed  limits  for  lateral  stresses.  The  details  of  the  pin  bearings  and  the  sizes  of 
the  pins  should  be  carefully  computed  by  the  same  methods  as  those  used  for  the  main  truss. 
The  attachment  to  the  truss  should  always  have  sufficient  rivets  to  take  the  maximum  chord 
increment,  which  equals  the  chord  component  of  the  largest  rod  at  the  point,  and  the  attach- 
ment to  the  strut  or  flange  of  the  floor-beam  should  have  sufficient  rivets  to  take  the  maximum 
shear  at  that  point.    These  details  will  be  left  to  the  student  to  proportion. 

Complete  detail  sketches  of  this  span  are  shown  in  Plate  I. 


HIGHWAY  BRIDGES. 


343 


CHAPTER  XXII. 
HIGHWAY  BRIDGES. 

346.  Definition  and  General  Remarks. — Under  this  general  classification  of  highway 
bridges  is  included  all  bridges  used  for  roadway  purposes  alone.  The  factors  which  control 
the  design  of  this  class  of  bridges  are  very  different  from  those  affecting  the  design  of  a  rail- 
way structure  and  make  it  well-nigh  impossible  to  treat  of  the  highway  bridge  in  general,  as 
each  structure  usually  presents  a  different  problem.  The  probable  maximum  live  loads  in 
quantity,  kind,  and  frequency  of  application,  the  expert  attention  which  the  bridge  will  re- 
ceive, the  ability  of  the  community  to  pay  for  a  bridge  of  a  capacity  beyond  their  present  needs, 
and,  in  some  cases,  the  appearance  of  the  bridge  when  completed  are  some  of  the  factors 
which  enter  into  the  problem  and  which  the  engineer  must  consider.  Usually  the  lightest 
structure  consistent  with  absolute  safety  and  one  which  will  require  little  or  no  e.xpert  atten- 
tion is  required.  The  economical  design  of  the  trusses  and  of  the  details  of  construction 
result  in  a  larger  percentage  of  saving  in  a  highway  than  they  do  in  a  railway  span  and  are 
therefore  of  supreme  importance.  Where  the  light  live  loads  and  consequently  light  trusses 
of  the  usual  highway  bridge  are  taken  into  consideration  it  will  be  seen  that  for  maximum 
economy  of  material  close,  careful,  and  intelligent  designing  is  necessary.  It  is  claimed  to 
the  credit  of  those  engineers  who  have  made  this  branch  of  bridge  designing  a  specialty  that 
the  highway  bridges  of  this  country  show  more  commendable  economy  of  design  than  do  thf 
railway  structures.  It  is  a  fact  that  our  highway  bridges  are  our  only  bridges  on  which  much 
effort  has  been  spent  to  make  the  design  neat  and  pleasing  in  outline. 

347.  Live  Loads. — The  live  loads  which  may  come  upon  a  highway  bridge  are  a  crowd 
of  people,  the  maximum  weight  resulting  from  a  closely  packed  throng  being  85  lbs.  per 
square  foot,  and  any  concentrated  load  due  to  the  passage  of  a  heavy  wagon  or  other  vehicle 
over  the  bridge.  In  estimating  the  live  load  resulting  from  a  crowd  of  people  the  probability 
and  also  the  possibility  of  a  closely  packed  crowd  covering  all  or  part  of  the  span  must  be 
taken  into  account.  A  good  rule  is  to  assume  a  live  load  from  this  cause  of  100  lbs.  per 
square  foot  for  spans  of  100  feet  and  under,  and  50  lbs.  per  square  foot  for  all  spans  over  250 
feet,  and  to  reduce  the  weight  per  square  foot  uniformly  from  lOO  to  50  lbs.  as  the  span  varies 
from  100  feet  to  250  feet.  Consistency  would  require  that  partial  loads  of  the  longer  spans 
be  increased  as  the  length  of  the  load  decreases  in  the  above  ratio.  The  above  specification 
would  be  for  bridges  subjected  to  city  traffic,  and  the  upper  limit  of  100  lbs.  for  a  span  of  100 
feet  may  be  reduced  for  bridges  in  localities  where  there  is  no  probability  of  such  dense 
crowds.  The  width  of  the  bridge  determines  to  some  extent  the  probability  of  there  being 
such  a  weight  of  people  on  it. 

The  concentrated  loads  are  estimated  for  the  special  locality  and  should  always  include 
probable  heavy  loads  which  the  development  of  the  adjacent  territory  may  make  necessary. 
There  is,  however,  a  great  waste  of  material  in  bridges  now  due  to  the  specification  of  heavy 
and  improbable,  if  not  impossible,  loads. 

348.  Dead  Loads. — The  fixed  or  dead  load  for  a  highway  span  consists  of  the  weight  of 
the  iron  in  the  span  and  of  the  floor  and  guards.  Owing  to  the  great  variety  of  floors  used 
and  the  differing  specifications  as  to  unit  stresses  and  details,  no  general  rule  easy  of  applica- 
tion has  yet  been  formulated  for  the  weight  of  iron  in  highway  bridge  spans  under  the  various 


344 


MODERN  FRAMED  STRUCTURES. 


conditions  and  specifications.  For  the  ordinary  truss  span  with  iron  trusses,  bracing,  and 
floor-beams  and  with  stringers  or  floor  joists  of  wood  the  weight  of  iron  per  foot  of  span  with 
a  live  load  capacity  of  1200  lbs.  per  linear  foot  is  closely  approximated  by  the  formula 

w  =  2l-\-  50, 

where  w  =  weight  of  iron  per  linear  foot,  and  /  =  the  length  of  the  span.  This  weight  does 
not  include  the  handrails.  Each  line  of  handrail  weighs  usually  about  25  lbs.  per  lineai  foot. 
The  weight  of  the  joists  and  flooring  can  be  determined  readily  from  the  sizes  used.  This 
formula  may  be  used  to  approximate  the  weight  of  iron  in  spans  of  a  capacity  greater  than 
1200  lbs.  per  fcot  by  increasing  the  weight  per  foot  in  the  same  ratio  as  the  capacity  is 
increased,  assuming  the  floor  to  be  as  before,  plank  laid  on  wooden  stringers  or  joists. 

349.  The  Various  Styles  of  Floors  Used  are  as  follows : 
1st.  Plank  in  one  or  two  layers  on  wooden  joists. 

2d.  Plank  laid  on  iron  joists  and  a  wearing  surface  of  wooden  blocks  used. 

3d.  Iron  joists  covered  with  corrugated  iron  or  buckle  plate  on  which  is  laid  a  bed  of 
concrete  to  receive  the  wearing  floor  of  asphalt,  granite  block,  or  vitrified  brick. 

The  joists  in  the  first  and  second  cases  are  spaced  from  two  to  three  feet  apart,  depending 
upon  the  thickness  of  the  plank  flooring  and  the  concentrated  loads  which  may  come  upon 
the  bridge.  For  the  third  style  of  floor  the  joists  are  spaced  as  far  apart  as  the  strength  of 
the  corrugated  iron  or  buckle  plate  flooring  will  permit.  The  standard  size  of  buckle  plates 
and  the  capacity  per  square  foot  for  different  thicknesses  may  be  obtained  from  the  manufac- 
turers on  request.  The  joists  for  this  style  are  usually  spaced  three  feet  or  more  centre  to 
centre. 

350.  Iron  Handrails  or  Fences. — The  ordinary  height  for  iron  handrailings  is  3  feet 
9  inches  from  the  floor  level  to  the  top  of  the  handrail.  They  should  be  stiff  enough  to 
resist  any  probable  force  which  would  tend  to  bend  them  out  of  line  or  knock  them  down. 
Nearly  all  the  standard  handrailings  used  on  bridges  will  fulfil  this  requirement  if  braced  at  dis- 
tances apart  of  about  8  feet.  The  lattice  work  should  be  made  so  that  the  openings  in  the 
fence  are  not  over  6  inches  square  for  the  lower  half  of  the  fence,  and  the  bottom  rail  should 
be  within  6  inches  of  the  floor  line. 

351.  The  Allowed  Unit  Stresses  for  Highway  Bridges  are  usually  25  per  cent 
higher  than  those  allowed  in  railroad  structures.  The  maximum -load  for  which  this  kind  of 
a  bridge  is  designed  is  usually  rarely  applied,  and  even  then  its  impact  is  not  as  destructive  as 
that  of  a  moving  train.  Wherever  the  stresses  from  the  assumed  loads  on  a  highway  bridge 
are  liable  to  occur  frequently  and  where  the  impact  is  appreciable,  as  in  the  case  of  street  cars 
for  example,  there  is  no  reason  why  the  allowed  stresses  should  differ  from  those  sanctioned 
by  railroad  engineers. 

352.  The  Details  of  Highway  Bridges  should  be  designed  with  the  same  care  and 
with  more  attention  paid  to  the  non-eccentric  connection  of  the  several  members  meeting  at 
a  joint  than  would  be  given  to  the  design  of  railway  bridge  details.  The  members  of  the 
truss  are  usually  smaller  and  the  margin  of  safety,  due  to  the  use  of  higher  unit  stresses,  is 
less,  so  that  the  secondary  stresses  caused  by  eccentric  connections  have  a  more  destruc. 
tive  effect  than  they  do  for  a  railway  bridge.  As  the  details  and  "  non-effective"  material 
such  as  tie  plates  and  lattice  bars  are  quite  a  large  percentage  of  the  material  in  a  span,  it 
is  commendable  economy  to  design  very  close  to  the  required  limits.  In  Chapter  XVIII  the 
customary  sizes  of  lattice  bars  and  the  spacing  of  the  same  are  given. 

353.  The  General  Dimensions  to  be  given  to  a  highway  bridge  are  usually  determined 
by  local  conditions.  The  width  of  roadway  is  usually  made  a  multiple  of  8  feet  for  each  car- 
riageway.   The  width  of  the  roadway  and  sidewalks  to  be  provided  depends  upon  the  traffic, 


HIGHWAY  BRIDGES. 


34S 


as  it  is  desirable  to  make  them  wide  enough  to  prevent  continual  crowding  during  those  hours 
of  the  day  when  the  traffic  is  the  greatest.  The  sidewalk  is  not  often  made  less  than  4  feet 
wide,  or  the  roadway  less  than  12  feet.  Wheel  guards  must  be  placed  on  the  floor  on  each 
side  of  the  roadway  and  so  located  that  when  the  wheel  of  a  vehicle  is  running  close  against 
it  the  hub  or  any  part  of  the  vehicle  will  not  strike  the  iron  work  of  the  trusses.  This  is 
generally  accomplished  when  the  guard  is  made  6  inches  wide. 

The  overhead  clearance  or  the  distance  from  the  top  of  the  floor  to  the  under  side  of  the 
top  lateral  bracing  for  through  bridges  should  be  14  feet  or  over  to  allow  high  loads  to  pass 
through.  A  farmer's  load  of  hay  will  pass  through  the  ordinary  barn  door,  which  is  12  feet 
high  in  the  clear. 

354.  The  Panel  Length  of  a  highway  bridge  in  which  wooden  joists  are  used  is 
limited  to  20  feet  or  less,  depending  on  the  maximum  concentrated  load  which  the  joist  has 
to  carry  and  the  size  of  joist  obtainable.  Joists  14  inches  high  are  common  sizes  and  are 
cheaper  than  those  of  greater  depth,  so  that  ordinarily  the  length  of  panel  is  limited  to  that 
which  is  the  maximum  span  over  which  this  size  will  carry  the  load.  For  bridges  with  iron 
joists  the  panel  length  can  be  varied  to  obtain  the  maximum  economy  in  iron  alone.  Where 
the  joists  are  supported  on  the  top  flange  of  the  floor-beam,  as  is  usual,  the  size  of  joist  to  use 
may  be  determined  by  the  permissible  fibre  stress  alone.  If  the  joists  arc  iron  and  placed 
between  the  beams,  like  the  stringers  of  the  span  designed  in  Chapter  XXI,  they  should  not 
be  less  than  one  fifteenth  of  the  span,  owing  to  the  excessive  deflection  of  shallower  joists 
loosening  the  rivets  in  the  connection  between  the  joist  and  the  floor-beam. 

355.  The  Kind  of  Construction  usually  employed  in  highway  bridge  construction  is 
I  beams  or  plate  girders  for  spans  under  30  feet  and  riveted  or  pin-connected  trusses  for 
longer  spans.  The  "  low  "  truss  or  half  through  bridge  is  employed  for  spans  of  70  feet  and 
under.  Where,  however,  a  deck  bridge  can  be  used  it  is  always  preferred  to  the  through,  as  it 
leaves  the  deck  or  floor  unobstructed. 

356.  The  Design  of  a  Highway  Span. — The  length  centre  to  centre  of  end  pins  will 
be  105  feet,  the  panel  length  15  feet,  and  the  depth  20  feet.  The  roadway  will  be  16  feet 
wide  and  the  two  sidewalks  5  feet  wide  in  the  clear.  The  live  load  will  be  assumed  at  2400 
lbs.  per  linear  foot  of  bridge.  A  concentrated  load  of  10,000  lbs.  on  two  axles  6  feet  apart 
will  also  be  provided  for.    The  allowed  unit  stresses  will  be  as  follows : 

Wrought-iron  in  tension,  12,000  lbs.  per  square  inch. 
Steel  in  tension,  15,000  "     "       "  " 

Wrought-iron  in  compression,  -  "^'^^^  lbs.  per  square  inch, 

where  /  =  length  of  member  in  inches,  g  =  least  radius  of  gyration  of  the  mem- 
ber in  inches,  and  a  —  36,000  for  two  flat  ends,  24,000  for  one  pin  and  one  flat 
end,  and  18,000  for  two  pin  ends. 
The  extreme  fibre  stress  on  steel  pins,  22,500  lbs.  per  square  inch. 
The  bearing  stress  on  pins  and  rivets,  15,000  "      "       "  " 
The  shearing  stress  on  pins  and  rivets,  9,000  "      "       "  " 
The  extreme  fibre  stress  on  yellow  pine  or  white  oak,  1200  lbs.  per  square  inch. 
For  stresses  in  the  lateral  system  increase  the  above  50  per  cent. 

357.  The  Design  of  the  Floor. — The  roadway  floor  will  be  assumed  to  be  three-inch 
white  oak  plank  laid  on  yellow  pine  joists.  The  three-inch  plank  will  be  laid  transversely,  and 
one  plank  10  inches  wide  will  carry  one  wheel  of  the  concentrated  load  30  inches  without 
exceeding  the  allowed  fibre  stress  on  white  oak.    Eight  joists  will  be  used,  spaced  x6  -i-  y  = 


346 


MODERN  FRAMED  STRUCTURES. 


2.29  feet  or  2  feet  3  inches  apart  centre  to  centre.  The  joists  will  be  figured  to  carry 
a  live  load  of  either  100  lbs.  per  square  foot  or  the  concentrated  load  specified.  Each  joist 
will  have  to  carry  one  quarter  of  the  total  weight  of  the  concentrated  load,  as  the  floor  plank 
are  stiff  enough  to  distribute  the  total  load  over  four  joists.  The  dead  load  on  each  joist  is 
12  lbs.  per  square  foot  for  the  floor  plank,  and  one  half  of  this  or  6  lbs.  per  square  foot  will  be 


assumed  for  the  joist.    The  dead  load  centre  moment  on  the  joist  is  then 


2^  X  18  X  15  X  180 


=  13,670  in.-lbs.      The  centre  moment  from  the  100  lbs.  per  square  foot  live  load  is 

2i  X  100  X  15  X  180  •     lu        TU  •  ^  c     '^  .    J  r 

—   =  75,940  in.-lbs.     Ihe  maxunum  moment  from  the  concentrated  live 

8 

load  is  1000  X  72*  =  72,000  in.-lbs.  The  100  lbs.  per  square  foot  live  load  produces  the 
greater  bending  moment.  The  total  maximum  bending  moment,  dead  and  live,  is  89,610  in.- 
lbs.,  requiring  a  joist  3"  X  13".  The  joists  are  usually  sawed  to  dimensions  in  even  inches, 
and  the  width  for  the  roadway  joist  should  never  be  tess  than  three  inches.  The  weight  of 
the  timber  in  the  roadway  floor,  assuming  the  weight  of  the  timber  to  be  4  lbs.  per  foot,  board 
measure,  is  as  follows  : 


Floor  plank  3"  X  12"— 16  ft.  long  =  48  ft.  B.M. 

Joists  eight  3"  X  13" —  i  ft.  long  =  26  ft.  B.M. 

Wheel  guards  two  6"  X  6"—  i  ft.  long  =   6  ft.  B.M. 

Hub  guards  two  3"  X  8"—  i  ft.  long  =  4  ft.  B.M. 


192  lbs. 

104  " 
24  " 
16  " 


Or  a  total  of  336  lbs.  per  linear  foot  of  bridge. 

The  sidewalk  floor  will  be  two-inch  plank  laid  on  three  joists  for  each  walk.  The  live 
load  will  be  taken  at  80  lbs.  per  square  foot.  The  dead  load  on  the  joists  is  8  lbs.  per 
square  foot  for  the  floor  plank  and  4  lbs.  per  square  foot  assumed  for  the  joist.    The  maxi- 

2^  X      X  I    X  180 

mum  bending  moment,  dead  and  live,  on  the  joist  is  —  —   =  77,600  in.-lbs., 

8 

requiring  a  joist  2^  X  12  inches.  The  weight  of  the  timber  in  the  sidewalk  floors  is  164  lbs. 
per  linear  foot  of  bridge.  The  sidewalk  floor  plank  are  usually  extended  to  close  the  open- 
ings between  the  roadway  and  sidewalks  made  by  the  trusses. 

358.  The  Design  of  the  Floor-beam. — The  dead  and  live  loads  on  the  beam  are  as 


■2.-5^«-3.25- 


 Live  Load  210(10  lbs— 

—Dead — '-^ —  88U0  Ibs.- 


II  I II 


-17.5- 


■>j^3.'25— j^2.'5 


Fig.  377. 

shown  in  Fig.  377.  The  maximum  bending  moment  occurs  at  the  middle  of  the  beam  where 
the  sidewalks  are  unloaded  and  is  64,300  ft.-lb.s.  The  depth  of  the  economical  floor-beam, 
using  a  web  plate  \  inch  thick  and  using  one  eighth  of  the  gross  area  of  the  web  as  available 
equivalent  flange  area,  is  20  inches  deep.  Assuming  the  depth  centre  to  centre  of  gravity 
of  the  flanges  as  18^  inches,  the  flange  stress  is  41,700  lbs.,  requiring  3.48  square  inches  net 
area  of  lower  flange  or  2.86  square  inches  net  area  in  the  flange  angles  after  deducting  the 
equivalent  flange  area  of  the  web.  Two  3"  X  3"  angles  weighing  17  lbs.  per  yard  will  be  used. 
The  top  flange  will  be  made  the  same  gross  area  as  the  bottom  flange.  The  maximum  bend- 
ing moment  from  the  overhanging  load  of  the  sidewalks  occurs  at  the  hanger  point  and  ia 
27,000  ft.-lbs.    The  flange  angles  and  web  plates  will  be  made  in  single  lengths  from  end  to 

*  See  Art.  91.  For  two  equal  loads,  placed  a  fixed  distance  d  apart,  the  maximum  moment  is  one  fourth  from  the 
centre  and  under  one  of  the  loads,  the  other  load  being  three-fourths  d  on  the  other  side  of  the  centre. 


HIGHWAY  BRIDGES. 


347 


end  of  the  beam.  The  load  on  the  hanger  supporting  the  floor-beam  from  the  truss  pin  is 
24,400  lbs.,  and  if  a  unit  stress  three  quarters  of  that  used  for  long  tension  members  or  flanges 
is  used  it  would  require  one  loop  hanger  i      inches  square. 

Referring  to  Plate  II,  it  will  be  noticed  that  the  lateral  rods  are  in  the  plane  of  the 
top  flange  of  the  beam,  and  that  this  flange  is  utilized  as  the  strut  for  the  lateral  system.  This 
flange  is  supported  laterally  every  two  feet  by  the  joists,  so  that  the  allowed  stress  from  the 
combination  of  live,  dead,  and  wind  stresses  may  be  as  much  as  15,000  lbs.  per  square  inch 
There  is,  then,  no  increase  of  the  section  of  the  flange  necessary.  The  floor-beams  are  riveted 
to  the  post  to  transfer  the  chord  component  of  the  lateral  rods.  The  top  flange  angles  are 
extended  to  take  the  brace  from  the  handrail.  Stiffener  or  distributing  angles  should  be 
riveted  to  the  web  of  the  beam  on  each  side  of  the  hanger  from  the  truss  pin  to  resist  the 
reaction  of  the  floor-beam.  This  end  reaction  is  24,400  lbs.,  requiring  24,400  -r-  2800  =  9 
|-inch  rivets  in  bearing  against  the  quarter-inch  web.  Four  angles  3"  X  2"  X  with  two 
vertical  lines  of  five  rivets  each  will  be  used  at  each  end  of  the  beam.  The  minimum  rivet 
spacing  through  the  flanges  allowable  is  4  inches,  and  is  computed  as  follows :  The  maximum 
shear  is  16,400  lbs.,  the  ratio  of  the  moment  of  resistance  of  the  flanges  to  that  of  the  web  is 
as  four  to  one ;  hence  the  amount  of  flange  stress  transferred  to  the  flange  is  four  fifths  of 

5r// 


what  it  would  be  if  the  web  were  neglected,  and  therefore  ^pS  =  r/i,  ox  p  — 


AS 


4.0.  In 


this  formula  p  —  rivet  pitch,  S  =  the  shear,  r  =  value  of  one  rivet  in  bearing  against  the  web 
plate,  and  /i  =  distance  between  the  rivet  lines  in  the  top  and  bottom  flanges.  The  rivets 
will  be  spaced  3  inches  apart  until  the  bearing  points  of  the  first  two  joists  are  passed,  when 
the  spacing  will  be  changed  to  6  inches,  using,  however,  two  3-inch  spaces  directly  under 
each  joist  bearing  to  avoid  the  use  of  distributing  angles. 

359.  The  Design  of  the  Trusses. — The  dead  load  will  be  assumed  as  2(2  X  105  -|-  50) 
=  520  lbs.  per  linear  foot  for  the  iron,  50  lbs.  per  hnear  foot  for  the  handrailing,  and  500  lbs. 


\ 

\  / 
\  / 

a       •  I 

< 

d 

f 

Member. 

Stresses. 

I 

g 

Area 
required. 

Dead. 

Live. 

Total. 

Unit 
stress. 

ab-c 

—  18.0 

-  40.5 

-  58.5 

150 

3-9 

cd 

-  30.0 

-  67.5 

-  97-5 

<< 

6.5 

de 

—  36-0 

—  81.0 

—  117. 0 

(  ( 

7.8 

Be 

—  20.0 

-  46.9 

-  66.9 

4-5 

Cd 

—  10. 0 

—  30.0 

—  40.0 

12.0 

3-3 

De 

0.0 

—  16.9 

—  16.9 

(( 

1.4 

Rf 

-f-  10. 0 

-  7-5 

0.0 

0.0 

Bh 

-  6.0 

—  i8.o 

—  24.0 

2.0 

Cc 

-f-  10. 0 

+  240 

-f  36-0 

7.0 

240" 

2.73" 

51 

Dd 

-|-  2.0 

+  13-5 

+   15  5 

5-5 

240 

2.0 

2.8 

aB 

+  30.0 

+  67.5 

+  97-5 

7.6 

300 

4.0 

12.8 

BC 

-|-  30.0 

+  67.5 

+  97-5 

9.0 

180 

3-4 

10.8 

CD 

4-  36-0 

-)-  81.0 

+  117.0 

9.2 

180 

3-3 

12.7 

DE 

+  36.0 

4-  81.0 

+  1170 

9.2 

180 

3-3 

12.7 

Make-up  of  Section. 


Two 
Two 
Two 
Two 
Two 
One 
One 
Two 
Two 
Two 
One 
Two 
Two 
Two 
Two 
Two 


2f'  X 

4"  X 
4"  X 

3"  X 

2i"  X 

Its"  s 
r  tie 
l"  squ 
7"  t-Js, 
5"  t-Js 
12"  X 

9"^ 
2V'  X 

9"  L  IS 
9"  LJ? 
9"  LJS 


'  bars 
bars 
bars 
bars 
bars 
quare  rod 
rod 

are  rod 

,  25A  lbs.  per  yd. 

18  lbs.  per  yd. 
\  top  plate 
s,  43  lbs.  per  yd. 
yV'  flats 
54  lbs.  per  yd. 
64  lbs.  per  yd. 
,  64  lbs.  per  yd. 


Area 


4.0 
6.5 
8.0 
4.5 

3-4 
1.4 

0.8 
2.0 
51 
3.6 

13-8 

10.8 
12.8 
12.8 


Material. 


Steel 


Iron 


The  stresses  in  the  table  are  given  in  thousand-pound  units.        denotes  compression:  —  denotes  tension. 


348 


MODERN  FRAMED  STRUCTURES. 


per  linear  foot  for  the  floor  and  joists,  making  a  total  of  1070  lbs.  per  linear  foot.  The  panel 
concentration  from  this  for  one  truss  is  ^(lo/o  X  15)  -  8000  lbs.  nearly.  Of  this  2000  lb. 
will  be  assumed  as  concentrated  at  the  top  chord  joints  and  6000  lbs.  at  the  bottom  chord 
joints.    The  live  load  is  2400  lbs.  per  linear  foot  or  18,000  lbs.  per  truss  panel 

The  tabl^on  page  347  gives  the  stresses,  required  areas,  and  make-up  of  the  sections. 

The  member  aB  was  made  as  shown  in  Fig.  378  in  order  to  use  a  top 
plate  to  transfer  the  wind  stress  down  the  post  and  at  the  same  time  keep 
the  pin  centres  in  the  middle  of  the  channel  webs.    The  sections  BC,  CD, 
and  DE  have  latticing  on  bo.h  their  top  and  bottom  flanges,  and  as  they 
s    are  perfectly  symmetrical  the  neutral  axis  is  in  the  centre  of  the  webs  of  the 
Fig.  378.  channels. 

360.  The  Wind  Bracing.-The  top  lateral  bracing  will  be  proportioned  to  resist  a  wind 
presssure  of  150  lbs.  per  linear  foot  of  the  bridge. 


Member. 

Stresses. 

i 

/ 

Area 

Make-up  of  Section. 

Wind. 

Unit  Stress. 

S 

required. 

Area. 

Material. 

BB' 
CC 
DD' 
BC 

ciy 

DE 

+  4 • 500 
-|-  2.250 

-  5-9 

-  30 
0.0 

7.0 
7.0 
18.00 
18.0 
18.0 

200 
200 

I. 15 

I-I5 

.64 
•32 
•33 
•17 
.00 

Four  3  '  X  2  '  X  i"  Ls 
Two  3"  LJS,  15  lbs.  per  yd. 
Two  3"  i_Js,  15  lbs.  per  yd. 
One  1 '  tie  rod 
One  1"  tie  rod 
One  1"  tie  rod 

4.8 

3-0 

30 
.60 
.60 
.60 

Iron 

The  bottom  lateral  bracing  will  be  proportioned  to  resist  a  static  wind  force  of  150  lbs. 
per  foot  of  bridge,  and  in  addition  a  moving  wind  force  of  100  lbs.  per  foot  of  bridge. 


a' 

b' 

c'  d' 

e' 

Member. 

Stresses. 

Area  required. 

Make-up  of  Section. 

Wind. 

Unit  Stress. 

Area. 

Material. 

ab' 

b^ 

cd' 

de' 

15.0 
9.2 
5-7 
r-5 

18.0 

•83 
•51 
•32 

.08 

One  iJ-g"  tie  rod 
One  1 '  lie  rod 
One        tie  rod 
One  I"  tie  rod 

.89 
.60 
.60 
.60 

Iron 

The  details  for  this  span  may  be  worked  out  by  the  student, 
are  given  in  Plate  II. 


Complete  detail  sketches 


THE  DETAIL  DESIGN  OF  A  HOWE  TRUSS  BRIDGE. 


349 


CHAPTER  XXIII. 
THE  DETAIL  DESIGN  OF  A  HOWE  TRUSS  BRIDGE. 

361.  The  Howe  Truss  has  proved  the  most  useful  style  of  bridge  ever  devised  for  use 
in  a  new  and  timbered  country.  It  is  still  very  largely  used  in  America  for  both  highway  and 
railway  purposes.  Railroad  bridges  of  this  kind  are  usually  built  on  the  ground  by  squads 
of  "  bridge  carpenters,"  and  are  very  cheap.  They  are  also  rigid  and  perfectly  safe  if  care- 
fully inspected  for  evidences  of  decay.  They  are  built  wholly  of  timber  except  the  vertical 
and  lateral  tie  rods  and  certain  joint  castings,  splicing  members,  bolts,  etc.  They  can  be  put 
together  in  such  a  way  that  any  single  stick  may  be  removed  and  replaced  while  in  service 
without  endangering  the  structure.  One  of  the  standard  Howe  truss  drawings  of  the 
Chicago,  Milwaukee,  and  St.  Paul  Railway  is  given  in  Plate  III,  and  the  bill  of  materials  at 
the  end  of  this  chapter.  This  plate  shows  a  span  147'  2\"  long,  composed  of  thirteen 
panels  of  10'  \\^'  at  the  bottom  chord  and  10'  \\\"  at  top  chord,  the  height,  inside  to 
inside  of  chords,  being  25'.  Similar  standard  plans  of  the  same  height  and  panel  length  are 
used  for  lengths,  diminishing  by  single  panels,  down  to  seven,  o"r  for  a  length  of  81'  6f"  centre 
to  centre  of  bottom  chord  joints.  All  these  standard  plans  and  the  corresponding  bills  of 
materials  are  lithographed  and  placed  in  the  hands  of  the  bridge  carpenters,  to  work  by.  All 
ends  are  square-sawed,  and  there  is  not  a  mortise  or  tenon  in  the  whole  structure. 

Sometimes  the  timber  bottom  chord  is  replaced  by  eyebars,  when  it  is  called  a  Combina- 
tion Bridge.  But  since  timber  is  much  stronger  in  tension  than  in  any  other  way,  there  is  no 
good  reason  for  doing  this. 

At  30  dollars  per  M,  the  cost  of  a  cubic  foot  of  timber,  in  a  bridge,  would  be  36  cents, 
while  the  cost  of  a  cubic  foot  of  iron  would  be  about  20  dollars.  The  working  stress  on 
timber  may  be  as  much  as  one  tenth  of  that  on  iron,  so  that  the  relative  first  cost  of  timber 
and  iron  structures  is  about  as  i  to  4  or  5. 

When  bridges  are  to  be  erected  far  from  existing  lines  of  railroad,  iron  bridges  become 
impracticable,  especially  when  timber  is  convenient. 

The  designing  of  a  Howe  truss  bridge  is  mostly  a  matter  of  joints  and  splicing.  The 
timber  sections  are  not  computed  with  that  care  that  is  used  in  iron  structures,  but  material 
is  used  liberally  and  usually  far  in  excess  of  what  the  formulas  would  give.  For  this  reason 
wooden  or  combination  bridges  should  not  be  let  out  by  contract  under  general  specifications, 
such  as  are  used  for  iron  bridges,  unless  all  the  sizes  are  specified  on  standard  typical  draw- 
ings like  that  of  Plate  III. 

By  referring  to  the  Table  of  Strength  of  Materials  at  the  end  of  the  volume,  it  will  be 
'seen  that  timber  is  relatively  very  weak  in  lateral  compression  and  in  shearing  along  the  grain. 
For  this  reason  a  timber  post  or  strut  should  never  bear  on  the  side  of  another  timber, 
but  always  on  a  wrought-  or  cast-iron  plate,  which  in  turn  presents  a  greatly  increased  surface 
to  the  lateral  face  of  the  wood. 

362.  Chord  Splices.— The  chords  are  made  up  of  as  long  timbers  as  possible,  and  but 
one  stick  spliced  at  any  section.  By  referring  to  Plate  III  and  to  the  bill  of  materials 
accompanying  it,  it  will  be  seen  that  the  standard  length  of  stick  in  the  upper  chord  is  43'  \\" 
(cut  from  44'  sticks)  and  in  lower  chord  it  is  615'  7^"  (cut  from  66'  sticks),  the  only  variation 
being  for  the  supplementary  end  sections.    The  chords  are  packed  twice  in  each  panel,  be- 


350 


MODERN  FRAMED  STRUCTURES. 


Oak 


6 
Pine 
6 


Oafc 


Pine 


Fig.  379, 


tween  the  angle-blocks,  and  the  splice  is  always  made  at  one  of  these  points.  The  timber  is 
always  sawed  for  these  bridges  on  special  orders,  and  sticks  of  these  lengths  can  be  obtained 
in  the  South,  West,  and  Northwest. 

TJie  upper  chord  splice  consists  simply  of  square  abutting  ends,  at  the  middle  of  pine 
packing  blocks  2\  inches  thick  and  10  inches  long,  dressed,  and  set  into  the  chord  sticks  |  inch 
on  each  side,  thus  leaving  a  |-inch  air-space  between  the  sticks.  When  the  outside  stick  is 
spliced  three  bolts  are  used  in  place  of  two,  as  shown  on  the  plate. 

The  lower  chord  splice  offers  more  of  a  problem.    It  was  formerly  the  custom  to  make 

this  splice  by  means  of  two  oak  fish-plates  notched 
into  the  sides  of  the  stick  as  shown  in  Fig.  379. 
There  were  five  ways  in  which  this  joint  might  fail. 
1st.  At  aa,  by  crushing  down  the  fibres. 
2d.  At  bb,  by  shearing  off  the  shoulders  of  the 
stick,  which  would  usually  be  white  pine. 

3d.  At  cc,  by  shearing  off  the  shoulders  of  the 
splice  plates. 

4th.  At  dd,  by  rupturing  the  splice  plates  in 
tension,  or  partly  in  shear  if  they  were  somewhat 
cross-grained. 
5th.  At  ee,  by  rupturing  the  main  stick  in  tension. 

These  methods  are  given  in  the  order  of  their  most  common  occurrence. 

Since  strength  against  rupturing  at  aa  is  only  obtained  by  sacrificing  that  at  ee,  it  is 
evident  that  the  greatest  strength  of  the  joint,  so  far  as  these  two  methods  of  failure  are  con, 
cerned,  is  obtained  when  we  have  notched  in  so  far  that  we  have  an  equal  strength  in  these 
two  ways.  The  endwise  crushing  strength  of  white  pine  is  not  less  than  4000  lbs.  per  square 
inch,  and  the  tensile  strength  is  not  less  than  7000  lbs.  per  square  inch.  To  realize  the 
greatest  strength  at  this  section,  therefore,  the  stick  should  be  notched  more  than  a  quarter  of 
its  width.  To  provide  for  the  case  of  cross-grained  wood  at  this  point,  it  is  a  good  rule  to 
notch  one  fourth  the  thickness  of  the  stick  on  each  side. 

The  shearing  strength  of  white  pine  is  only  about  400  lbs.,  or  say  one  tenth  of  the 
strength  in  crushing  endwise.  Hence  the  section  in  shear  at  bb  should  be  ten  times  the 
depth  of  the  notch,  or  2^  times  the  thickness  of  the  stick.  If  the  splicing  timbers  are  also  of 
white  pine  the  above  results  apply  to  them  too,  but  if  they  are  of  selected,  straight-grained, 
seasoned  oak,  or  better  of  long-leaf  yellow  pine  {pinus  pahistris),  then  the  thin  portions  at  dd 
need  be  only  about  one  sixth  or  one  eighth  of  the  thickness  of  the  main  timber,  and  the  length 
of  the  shoulder,  cc,  only  one  half  that  on  the  white  pine  timber  itself  at  bb.  Thus  for  splicing 
a  white  pine  stick  8  inches  thick  with  white  pine  splices,  the  joint  would  be  80  inches  long 
and  the  splices  made  of  4-inch  stuff,  notched  out  2  inches.  If  made  of  long-leaf  yellow  pine  or 
of  straight-grained  oak,  the  joint  would  be  60  inches  long,  made  of  3^-inch  stuff  and  notched 
out  2  inches,  leaving      inches  thickness  at  dd. 

No  reliance  should  ever  be  put  upon  the  bolts.  They  serve  simply  to  hold  the  parts  together, 
and  would  not  come  into  action  at  all  until  there  had  been  considerable  movement,  and  then 
they  would  act  very  imperfectly  and  to  an  unknown  extent.  A  good  joint  will  develop  its  full 
working  strength  without  any  appreciable  distortion. 

The  great  length  of  these  timber  splices,  and  the  uncertainty  arising  from  imperfect 
workmanship  and  from  cross-grained  material,  which  is  apt  to  be  more  or  less  wind-shaken 
or  season-checked,  so  as  to  offer  little  resistance  to  shearing  along  the  grain,  has  resulted  in 
an  entirely  new  kind  of  tension  splice  now  adopted  on  many  of  our  leading  railways  of  the 
West. 

The  iron  splice  is  composed  of  two  cast-iron  plates  on  each  side  of  the  sticks  to  be  joined, 


THE  DETAIL  DESIGN  OE  A  HOWE  TRUSS  BRIDGE. 


351 


with  short  cylindrical  spurs  fitting  into  corresponding  bored  holes  in  the  stick,  these  being 
held  together  by  two  clamp  bars  of  wrought-iron,  having  forged  hooks  at  their  ends,  all  as 
shown  in  Plate  III.  The  cast-iron  plates  (and  hence  the  sticks)  are  drawn  tightly  together 
by  means  of  a  clamp  key  which  is  driven  at  one  end  of  each  clamp  bar,  the  key-seat  on  the 
cast-iron  plate  being  inclined  to  the  plane  of  the  key  as  shown.  Each  clamp  plate  here  shown 
has  twenty-one  spurs,  i  inch  long  and  \\  inches  diameter,  making  a  total  area  of  23!  square 
inches  bearing  area  for  each  plate,  or  47^  square  inches  for  each  stick.  The  area  of  the  cross- 
section  of  these  sticks  is  96  inches,  so  that  the  compression  bearing  area  is  about  one  half 
the  section  of  the  stick.  But  since  this  area  is  not  all  cut  out  at  the  same  section,  this  bearing 
area  might  well  be  increased,  even  to  the  extent  of  making  the  spurs  if  inches  long,  thus 
making  the  bearing  area  for  one  stick  74  square  inches. 

The  clamp  bar  has  a  minimum  section  of  3  square  inches,  but  with  longer  spurs  it  might  be 
widened  to  4  inches  and  its  section  made  4  square  inches.  This  would  give  8  square  inches 
of  section  of  bar  to  74  square  inches  of  bearing  area  on  the  wood,  or  9  square  inches  bearing 
to  I  square  inch  of  iron  in  tension,  which  is  about  the  proper  ratio. 

This  arrangement  of  iron  clamp  blocks  and  bars,  with  tightening  key,  is  a  most  excellent 
one  and  should  entirely  replace  the  old  wooden  fish-plates.  By  driving  out  the  key,  and 
slightly  spreading  the  chord  members,  any  single  stick  can  be  taken  out  and  replaced  by  a 
new  one. 

"363.  The  Angle  Blocks  and  Bearing  Plates. — Since  timber  offers  very  slight  resist 
ance  to  crushing  across  the  grain,  it  is  necessary  to  give  to  all  the  web  struts  and  vertical 
tension  rods  very  large  bearings  on  the  chords,  at  both  top  and  bottom,  in  order  to  prevent 
the  crushing  down  of  the  fibres.  The  resistance  of  white  pine  to  crushing  across  the  grain  is 
only  about  500  lbs.  per  square  inch,  and  of  yellow  pine  from  800  to  1000  lbs.  The  working 
stress  should  not  be  over  300  lbs.  for  white  pine  and  500  lbs.  for  yellow  pine. 

The  angle  blocks  shown  in  Plate  III  are  all  18  inches  wide  and  of  a  length  equal  to  the 
total  width  of  the  chords.  The  top  and  bottom  blocks  are  exactly  alike.  Two  ribs  project 
\\  inches  from  the  lower  faces,  which  are  notched  into  the  chords  to  take  the  lateral  thrust 
of  the  struts,  and  with  the  patterns  here  shown  the  vertical  components  of  the  thrust  are 
transmitted  through  the  chords  by  lateral  compression  on  the  timbers.  This  is  well  enough 
if  the  bearing  plates  at  top  and  bottom  are  of  sufficient  size,  which  in  this  design  they  are. 

The  bearing  plaies  diXQ  h.&xQ  composed  of  10-  and  12-inch  channel  bars  the  full  width  of 
the  chord  sections.  Thus  in  the  147-foot  span  shown  in  Plate  III  the  load  on  the  end  ver- 
ticals may  be  taken  at  150,000  lbs.  for  each  truss.  The  bearing  plate  here  is  12"  X  43"  = 
516  square  inches.  At  300  lbs.  per  square  inch  this  would  carry  a  load  of  154,800  lbs.,  which 
shows  it  to  be  of  sufficient  size.    Being  made  of  channel  iron,  it  is  presumed  to  be  stifif 


Fig.  380. 


enough,  if  large  washers  are  used  under  the  nuts,  to  spread  the  load  uniformly  over  the  bear- 
ing area. 

Another  method  of  transmitting  this  thrust  through  the  chord  section  is  to  extend  the 


352 


MODERN  FRAMED  STRUCTURES. 


web  of  the  angle  blocks  entirely  through  the  chords  to  receive  the  bearing  from  the  nuts 
directly.  This  enables  this  load  to  be  transmitted  wholly  through  the  cast  angle  block  and 
allows  no  lateral  compression  to  come  upon  the  timber  chords.  In  this  case  the  shrink- 
age of  the  bottom  chord  timbers  would  allow  the  angle  block  to  stand  off  from  its  seat  some- 
what and  make  a  good  reservoir  for  rain-water,  which  would  rapidly  rot  the  timber  at  a  point 
where  it  could  not  be  inspected.  A  close  bearing  on  cast-iron  makes  a  very  lasting  joint.  It 
/vould  seem,  therefore,  that  the  joint  shown  in  Plate  III  is  the  best  which  could  be  devised. 

364.  Miscellaneous  Details. — Dowels. — All  the  web  struts  are  held  in  place  by  iron 
dowel  pins  ^  inch  in  diameter  by  9  inches  long,  fitting  tight  into  bored  holes  in  the  ends  of  the 
struts  and  entering  corresponding  holes  in  the  angle  blocks.  They  simply  prevent  the  timbers 
from  slipping  out  of  place.  All  bridge  timbers  are  apt  to  be  more  or  less  green  when  put  into 
place,  and  although  most  of  the  shrinkage  is  in  a  circumferential  direction,  there  is  some 
shrinkage  lengthwise.  All  Howe  truss  bridges,  therefore,  should  be  tightened  up  frequently 
by  screwing  up  the  nuts  on  the  vertical  rods.  If  these  are  not  kept  tight  the  counter-struts 
will  become  loose  and  slip  off  their  seats  if  not  held  to  place  by  dowel  pins  or  otherwise. 

The  lateral  wind  bracing  consists  of  diagonal  timber  struts  and  transverse  iron  tie  rods, 
in  the  planes  of  both  top  and  bottom  chords.  The  angle  blocks  for  these  lateral  systems  are 
shown  in  Plate  III.  Instead  of  dowels  there  are  projecting  flanges  on  the  lower  sides  of  the 
angle  blocks  which  hold  the  diagonal  struts  in  place. 

Washers  and  Packing. — The  upper  chord  is  packed  with  seasoned  pine  blocks  except  at 
the  lateral  rods,  where  |^-inch  cast  washers  {P)  are  used.  The  lower  chord  is  packed  with  the 
cast  washer  F  except  along  the  lateral  truss  rod,  where  the  plane  washer  P  is  used  the  same 
as  in  the  upper  chord.  This  F  washer  is  intended  to  act  so  as  to  transfer  tensile  stress  from 
one  stick  to  another,  and  in  this  way  cause  the  chord  to  act  together  as  one  solid  stick.  For 
this  purpose  these  washers  are  rimmed,  and  these  rims  are  set  I  inch  into  the  timber  on  each 
side.  The  sockets  for  these  circular  rims  are  cut  out  with  a  special  tool,  truly  circular,  and 
the  projecting  rims  of  the  washer  are  given  a  "  draw  "  of  -^^  inch  on  each  side,  so  that  when 
the  packing  bolts  are  tightly  drawn,  these  rims  are  set  snugly  into  the  wood  on  either  side,  to 
a  depth  of  I  inch,  on  an  outside  diameter  of  6  inches,  there  being  two  of  tliese  at  each  pack- 
ing section.  These  sections  occur  every  5^  feet,  so  that  every  feet  a  common  union  of  12 
square  inches  area  is  made  between  each  pair  of  adjacent  sticks.  It  is-  this  transfer  of  stress 
laterally  from  one  stick  to  another  which  causes  the  total  weakening  effect  of  the  bottom 
chord  splices  to  be  limited  to  that  due  to  the  splicing  of  one  stick.  Thus  if  stick  two  has 
been  spliced  at  a  sacrifice  of  half  its  strength,  and  li  feet  farther  along  stick  three  is  to  be 
spliced,  there  will  be  four  of  these  rimmed  6-inch  washers  introduced  on  each  side  of  stick 
three,  after  passing  the  splice  in  two,  and  these  will  transfer  stress  from  stick  three  to  sticks 
two  and  four,  with  an  equivalent  area  of  48  square  inches  of  timber,  leaving  only  about  a  half 
load  in  three  to  be  carried  past  the  joint  over  the  two  sets  of  splice  bars  and  blocks.  This 
rimmed  washer  is  a  much  more  efficient  packing  and  transfer  block  than  the  solid  oak  packing 
blocks  notched  into  the  adjacent  timbers  which  were  formerly  common. 

Lip  Washers. — To  further  assist  in  holding  the  end  angle  blocks  to  place  cast-iron  lip 
washers  are  introduced  (pattern  I)  which  ?:avc  a  lip  projecting  over  the  free  edge  of  the 
block,  and  are  bolted  through  the  chord.  Each  end  block  on  the  bottom  chord  has  four  of 
these  back  of  it,  the  next  one  three,  and  the  third  from  the  end  two.  A  few  are  used  on  the 
end  upper  chord  blocks  also.  These  washers  also  hold  down  this  side  of  the  block  and  so 
assist  in  distributing  the  load,  which  comes  wholly  on  the  opposite  slope  of  the  block,  evenly 
over  the  whole  base  of  the  block. 

Collision  Struts  are  introduced  to  stay  the  end  inclined  struts  in  case  they  should  be  struck 
by  a  derailed  car,  engine,  or  projecting  timber. 

Portal  Braces  6"  X  8"  are  used  as  shown  in  Plate  III,  being  well  attached  both  to  the 


THE  DETAIL  DESIGN  OF  A  HOWE  TRUSS  BRIDGE. 


353 


inclined  end  posts  and  to  a  portal  strut  at  top  lo"  X  12",  which  in  turn  is  bolted  and 
shouldered  upon  the  ends  of  the  top  chord.  The  absence  of  any  very  efificient  wind,  sway,  or 
portal  bracing  (and  formerly  the  total  absence  of  any  such  bracing  whatever)  has  been  the 
cause  of  many  failures  of  Howe  truss  bridges,  and  is  still  an  objection  to  them.  It  has  been 
very  common  to  cover  in  Howe  truss  bridges  on  highways,  and  when  this  is  done  and  all 
portal  or  sway  bracing  omitted,  a  comparatively  small  wind  pressure  would  wreck  the  struc- 
ture. 

The  Floor-beams  and  Stringers  are  of  timber  and  usually  rest  directly  upon  the  bottom 
chord.  Sometimes  they  are  hung  below  it,  as  shown  in  Plate  III.  In  this  case  a  large  bearing 
area  must  be  provided  both  on  top  of  the  chord  and  at  the  base  of  the  beam,  to  prevent  the 
cutting  in  of  the  washers.  The  stringers  are  also  of  timber,  held  well  apart  by  means  of  pack- 
ing washers  or  thimbles,  three  inches  long,  thus  giving  not  only  a  free  circulation  of  air,  but 
allowing  live  coals  from  the  engine  an  opportunity  to  roll  off  and  drop  between  them.  The 
remaining  portion  of  the  floor  system  is  not  peculiar  to  this  style  of  truss. 

Corbels  are  usually  introduced  under  the  ends  so  as  to  save  the  bottom  chord  timbers  and 
keep  them  away  from  the  sills  or  wall  plates.  They  are  packed  with  the  plain  washers  and 
bolted  to  the  bottom  chord. 

365.  Working  Stresses. — As  stated  in  Art.  361,  the  timber  members  of  a  Howe  truss 
are  not  usually  nicely  computed  and  dimensioned  as  is  always  the  case  with  wrought  iron  and 
steel,  but  certain  maximum  loads  should  not  be  exceeded. 

Prof.  Lanza,  who  is  now  the  leading  authority  on  the  strength  of  timber,  gives  as  the 
results  of  actual  tests  on  large  beams  and  columns  the  following:* 

Working  Fibre  Stress  in  Cross-breaking,  using  a  Factor  of  Safety  of  Four,  on  Actual  Tests 
of  Full-sized  Beams. 

White  pine  and  spruce   750  lbs.  per  square  inch. 

Georgia  (long-leaf)  yellow  pine   1200   "     "       "  " 

White  oak   1000   "  " 


Columns. — For  the  ultimate  strength  of  timber  columns  use  the  formulas  given  on  p.  151, 
which. are  based  directly  on  Lanza's  published  results  of  full-sized  column  tests,  A  factor  of 
safety  of  four  or  five  should  be  sufficient. 

In  applying  the  formulae  to  composite  columns  made  up  of  several  sticks  bolted  to- 
gether at  intervals,  give  to  each  stick  its  proportionate  share  of  the  total  load  to  be  carried 
over  that  member,  and  then  assume  that  it  stands  alone  and  unsupported.  This  is  the  only 
safe  rule.  Even  though  they  are  firmly  bolted,  with  packing  blocks  or  washers  notched  into 
the  sides,  these  grow  loose  in  time  and  do  not  resist  initial  lateral  bending.  They  should  never 
be  assumed  to  act  as  one  solid  stick. 

Stiearing.\ — For  shearing  and  crushing  it  is  probably  safe  to  use  a  factor  of  safety  of 
two  or  three.  Pending  the  final  reports  on  the  U.  S.  Timber  Tests  now  in  progress,  use  for 
working  values  of  the  shearing  strength  along  the  grain,  for 

White  pine  

Long-leaf  yellow  pine 
Short-leaf     "  " 
White  oak  

*  Applied  Mechanics,  Fourth  Ed.,  pp.  670,  677,  684,  and  685. 

f  These  values  of  working  stresses  in  shearing,  crushing,  and  tension  cire  taken  from  the  results  of  the  U.  S. 
Timber  Tests  now  being  conducted  at  Washington  University  by  Prof.  Johnson  for  the  Forestry  Division  of  the 
Agricultural  Department. 


100  lbs.  per  square  inch. 
200  "      "       "  " 
150  "      "       "  " 
300  "      "       "  " 


354 


MODERN  FRAMED  STRUCTURES. 


Crushing  across  Grain. 

Take  for  working  values,  on  seasoned  timber,  for 

White  pine     300  lbs.  per  square  inch. 

Long-leaf  yellow  pine   500  "      "       "  " 

Short-leaf      "        "    450  "     "       "  " 

White  oak   1000  "     "       "  " 

Crushing  Endwise  {Short  Blocks). 

Take  for  working  stress  : 

White  pine    2500  lbs.  per  square  inch. 

Long-leaf  yellow  pine   3000  "      "       "  " 

Short-leaf     "        "    2800  " 

White  oak   2750  "      "       "  " 

Tension. — Wood  fibre  is  much  stronger  in  tension  than  in  any  other  way,  and  as  a  result 
it  may  be  said  that  wood  seldom  or  never  breaks  in  pure  tension  in  actual  service.  In  fact, 
it  is  very  difficult  to  break  it  in  tension  in  a  laboratory  test.  In  structures  timber  usually  fails 
in  shearing,  in  cross-breaking,  or  in  crushing.  It  must  always  be  assumed,  in  long  sticks  in 
tension,  as  in  the  bottom  chord  of  a  Howe  truss,  that  the  grain  runs  more  or  less  across  the 
line  of  the  stick,  and  a  liberal  allowance  must  be  made  for  the  reduction  of  the  section  by 
framing  it,  so  that  although  the  tensile  strength  of  the  fibre  maybe  ten  (or  in  the  case  of  long- 
leaf  yellow  pine  nearly  twenty)  thousand  pounds  per  square  inch,  yet  it  is  not  wise  to  rely  on 
a  working  stress  in  tension  of  more  than  about  1000  or  2000  lbs.  per  square  inch. 

366.  Weights  and  Quantities. — The  following  table  of  loads,  quantities,  and  weights  is 
taken  from  a  complete  scheme  of  stress  diagrams  and  sizes  for  both  deck  and  through  Howe 
truss  bridges  from  30  feet  to  1 50  feet  in  length,  for  the  Oregon  Pacific  Railway  (A.  A.  Schenck, 
Chief  Engineer),  published  in  Engineering  News,  April  26,  1890.  The  live  load  assumed  was 
two  88-ton  engines  followed  by  a  train  load  of  3000  lbs.  per  foot.  For  deck  bridges  add 
20  per  cent  to  the  weight  of  the  timber  and  deduct  20  per  cent  fxom  the  weight  of  the 
wrought-iron. 

WEIGHTS  AND  QUANTITIES  FOR  HOWE  TRUSS  BRIDGES. 


Assumed  Loading  per  Foot. 

Estimated  Quantities. 

Total 

Length  of 

Style  of 

Height  of 

No.  of 

Load  per 

Wrought-iron. 

Span. 

Truss. 

Truss. 

Panels. 

Foot. 
Lbs. 

Trusses. 

Floor. 

Train. 

Timber. 

Cast-iron, 

Lbs. 

Lbs. 

Lbs. 

Feet  B.  M. 

Rods  not 
Upset. 
Lbs. 

Rods 
Upset. 
Lbs. 

Lbs. 

30  feet 

Pony 

9  feet 

4 

360 

500  . 

5,060 

5.920 

10,160 

2,170 

970 

40  " 

( ( 

11  " 

4 

400 

500 

4,600 

5,500 

13,360 

2, 960 

1,210 

50  " 

(f 

II  " 

6 

450 

500 

4,200 

5.150 

lg,020 

5.610 

2,880 

60  " 

12  " 

6 

540 

500 

3,860 

4,900 

22,780 

6,790 

3,660 

70  •' 

<( 

13  " 

7 

600 

500 

3,640 

4.740 

29,930 

9,260 

8,210 

8,260 

80  ■' 

<  ( 

14  " 

8 

620 

500 

3,600 

4,720 

35,390 

11,660 

10,260 

9,970 

90  " 

15  " 

9 

720 

500 

3.560 

4,780 

42,710 

15,170 

13.440 

12,530 

90  " 

Through 

25  " 

8 

720 

500 

3.560 

4,780 

41,880 

17,880 

15,150 

12,260 

100  " 

25  " 

9 

800 

500 

3,500 

4,800 

48,890 

22,580 

18,950 

14.290 

IIO  " 

25  " 

10 

880 

500 

3,400 

4,780 

54,770 

25,820 

22.290 

•  5.930 

120  " 

25  " 

1 1 

940 

500 

3,300 

4.740 

62,040 

30,890 

26,010 

18,290 

130  " 

<( 

25  " 

12 

1,000 

500 

3,200 

4,700 

70,130 

37,050 

30,180 

20,830 

140  " 

25  " 

13 

1,050 

500 

3.150 

4,700 

78,160 

40,820 

33,020 

23.210 

150 

(f 

25  " 

14 

1,100 

500 

3,100 

4,700 

86,630 

48,090 

39,140 

27,060 

THE  DETAIL  DESIGN  OF  A  HOWE  TRUSS  BRIDGE. 


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356 


MODERN  I' RAM  En  STRUCTURES. 


367.  Long  Span  Howe  Truss  Bridges.— In  Plate  IIIa  are  shown  the  details  of  a  Howe 
truss  bridge  of  250  feet  span,  designed  and  built  for  the  Oregon  division  of  the  Southern 
Pacific  Railway  by  Mr.  W.  A.  Grondahl,  Resident  Engineer.  These  bridges  are  constructed 
fronn  "  Oregon  fir"  timber,  it  being  practicable  to  obtain  large  sticks  of  this  wood  as  long  as 
70  feet.  This  enables  a  new  departure  to  be  made  in  wooden  bridges,  and  Mr.  Grondahl  has 
solved  the  problem  in  an  eminently  satisfactory  manner.  On  the  Pacific  coast  this  timber 
is  cheap,  while  iron  is  very  dear,  and  hence  it  proves  to  be  economy  to  build  of  the  cheaper 
but  shorter-lived  material. 

The  drawings  given  in  Plate  IIIa  very  clearly  indicate  the  method  employed.  The  live 
load  for  which  the  bridge  was  designed  consisted  of  Cooper's  Class  Extra  Heavy  A  (see  p. 
79),  or  of  two  100-ton  engines,  followed  by  a  train  load  of  3000  lbs.  per  foot. 

The  peculiar  features  of  this  new  Howe  truss  are: 

First.  Its  great  height  (45  ft.  out  to  out),  and  its  long  panels  (31  ft.  3  in.),  which  are, 
however,  subdivided  in  the  lower  chord,  as  in  the  Baltimore  truss.  Fig.  97,  Art.  78. 

Second.  The  use  of  eyebars  and  large  steel  pins  in  making  the  bottom  chord  splices. 

Third.  The  use  of  timbers  10  in.  X  21  in.  X  63  ft.  long  in  making  up  the  chord  members, 
and  of  19  in.  X  20  in.  X  50  ft.  long  in  the  web  system. 

Fourth.  The  omission  of  all  counter-braces  where  they  are  not  required  by  the  analysis. 

In  the  working  out  of  this  design  the  following  features  should  be  noted  : 

The  Suspension  Joint  in  the  middle  of  the  web  braces,  as  .shown  in  Figs.  6,  7,  and  8.  The 
suspension  stirrups  hang  from  a  spool  7  in.  in  diameter,  on  the  top  of  which  rests  a  small 
strut  for  sustaining  the  weight  of  the  upper  chord,  which  member  is  held  in  place  by  an 
inverted  stirrup  passing  under  the  .spool.  The  spool  is  held  in  place  by  the  inclined  parts,  into 
which  it  is  seated  to  a  depth  of  6  inches  at  each  end,  as  shown  in  Fig.  6. 

The  Bottom  Chord  Splice  2ls  shown  in  Figs.  9  to  15.  Two  eyebars,  from  five  to  eight  feet 
long,  are  laid  beside  the  member  at  the  splice,  and  3^-inch  pins  used,  which  make  a  snug  fit 
in  the  timber.  To  prevent  these  pins  from  bending,  and  to  increase  the  bearing  area  on  the 
wood,  other  pins  are  inserted  in  front  of  the  main  pin,  and  cast-iron  plates,  one  inch  thick 
(Figs.  14  and  15),  set  in  so  as  to  make  a  close  fit,  thus  distributing  the  pull  on  the  eyebars 
upon  three,  four,  five,  or  six  pins,  as  the  pull  increases  from  the  end  towards  the  centre,  as 
shown  in  Figs.  10,  11,  12,  and  13.  After  these  plates  and  piiis  are  all  in  place  the  whole  com- 
bination is  tightened  up  hy  turning  an  eccentric  collar  on  the  pin  through  one  end  of  the  eye- 
bars,  as  shown  in  Fig.  9.  The  small  circles  there  shown  are  the  recesses  into  which  are  fitted 
the  lugs  of  the  spanner  used  to  turn  these  collars  180°,  thus  drawing  the  end  pins  about  ^  inch 
nearer  together.  This  takes  the  place  of  the  "  clamp  key "  used  in  the  C,  M.  &  St.  P. 
designs,  Plate  III.  The  ends  of  the  spliced  sticks  are  held  in  place  by  a  one-inch  bolt  and 
washers  placed  vertically  in  the  joint,  as  shown  in  Fig.  lO. 

The  Floor-beams  are  bunched  as  near  to  the  bottom  chord  joints  as  possible,  three 
10"  X  18"  sticks  making  one  beam. 

The  Transmission  of  Loads  through  the  Bottom  Chords  is  effected  by  means  of  cast-iron 
"  false  tubes,"  shown  in  Figs.  21  and  22.  These  bear  on  the  cast-iron  angle  blocks  above  and 
on  the  gib  plates  which  receive  the  nuts  and  washers  below.  They  are  cast  to  a  bevel,  or  slope, 
so  that  they  can  be  driven,  or  drawn  into  a  tight  fit  in  the  wedge-shaped  slots  cut  in  the  sides 
of  the  sticks  (Figs.  17,  18,  19,  and  20),  these  latter  being  held  laterally  by  two  |-inch  bolts. 

These  and  other  features  of  this  truss,  of  less  significance,  have  so  far  extended  the 
capacities  of  the  Howe  truss  as  to  open  up  a  new  field  of  application  for  it  in  the  Northwest 
or  wherever  such  timbers  as  here  described  can  be  obtained.  Recent  tests  of  this  timber 
made  by  Prof.  Johnson  show  it  to  be  superior  to  white  pine  in  strength  and  stiffness.  It  is 
of  a  straight  and  even  grain  free  from  knots,  wind  shakes,  and  season  checks,  and  promises  to 
become  the  most  valuable  structural  timber  in  the  world. 


Plate  III. 


f  JilU|iLJ||L^|iiJiiU|!LJ||--i 

 J.;  L!  a....-Li — J\. — : 


I       1  L      LL.  ..il  LL— — ll.   • 


-88- 


iiini 


CLAMP  BLOCK-RIGHT 

Pattern  R|. 
This  is  Pattern  K-ChnnKed 


h  lOi" 


.  iTTi .  T  nj'  aiii.of  Holf 


^It'-        ]      I       End  Hoivs  in  this  (.;hanlicl  to  l»e  Drillfd 

18 "iron  Channel  43 "long  X  Web. 


^  4- 

U  

^85— V 
 a 

12"lrou  Channel  30"long  5^"web. 


S«'»  e  y  10,13  4  ISS"  T«l-k.  UUw  fur  d;».orHol«. 

- — --^ 


 34-  


10""lron  Channel  34""long  x  'vVeb. 


.  124^  . 

.f,    .,,      r.   1.  „ 

All  contacts  of  wood  and  wood  to  be  tliorouj^hly  painted 
with  white  lead. 

All  bolts  ^"diameter  except  as  noted  for  suspended  floor 
Standard  slot  washers  for  %  bolt. 


STANDARD  HOWE  TRUSS 

—  ■  OF  THE  :— 

C.  M.  &    ST.  P.  RY. 

BRIDGE  &  BUILDING  DEPARTMENT 

147-2M"  HOWE  TRUSS  SPAN 
Chicago,  Illinois.  April.  ISfll. 


THE  DESIGN  OF  SWING  BRIDGES.  357 


CHAPTER  XXIV. 


THE  DESIGN  OF  SWING  BRIDGES. 


368.  Different  Types  of  Swing  Bridges  and  Determining  Conditions. — Swing 
bridges  are  known  generally  as  Centre-bearing  and  Rim-bearing,  according  to  the  manner  in 
which  the  bridge,  when  swinging,  is  carried  at  the  centre  pier.  If  the  entire  dead  load,  when 
swinging,  is  carried  on  a  vertical  pin  or  pivot,  it  is  called  Centre-bearing.  If  the  entire  dead 
load,  when  swinging,  is  carried  on  a  circular  girder,  called  a  drum,  which  in  turning  moves 
upon  rollers,  it  is  called  Rim  bearing.  Again,  these  two  conditions  are  combined  into  one,  so 
as  to  make  the  bridge  partly  rim-bearing  and  partly  centre-bearing.  The  conditions  which 
generally  determine  the  style  of  bearing  to  be  used  in  any  given  case  are  usually  concomitant 
with  the  style  of  the  bridge  proper  to  be  used.  In  general,  plate  girder  swing  bridges  have  a 
centre  bearing,  and  truss  bridges  have  a  rim  bearing,  or  a  rim  bearing  and  centre  bearing  com- 
bined. However,  in  practice,  the  condition  which  may  finally  determine  the  style  of  bearing 
to  be  used  is  the  vertical  height  available  under  the  bridge,  or  the  depth  from  the  base  of  rail 
to  the  top  of  pier.  This  distance  is  usually  limited,  and  compels  the  designer  to  resort  to 
various  devices  to  properly  carry  the  bridge  when  swinging. 

369.  Plate  Girder  Swing  Bridges. — These  should  be  used  for  all  lengths  up  to  100 
feet.*    For  lengths  from  100  to  160  feet  the  riveted  truss  design  is  preferable. 

The  joint  at  the  centre  for  plate  girder  bridges  is  usually  centre  bearing.  Fig.  381 
shows  the  cross-section  taken  near  the  centre  of  a  plate  girder  swing  bridge.  The  main 
girders  A,  A,  for  a  through  bridge,  are  usually  spaced 
14  feet  apart.  The  part  marked  C  is  the  centre  cast- 
ing, which  receives  the  centre  pin  or  pivot  P,  upon 
which  the  cross-girders  rest.  The  parts  marked  B  and 
B  are  supports  for  the  main  girders  A  and  A  when  the 
bridge  is  closed.  These  supports  relieve  the  pivot 
from  carrying  any  live  load,  except  such  as  comes 
directly  on  the  cross-girders.  When  the  bridge  is 
opened,  the  supports  B  and  B  being  fi.xed  to  the  ma- 
sonry, the  entire  weight  is  carried  on  the  pivot  P.  In 
designing  the  pivot,  care  should  be  exercised  to  have 
the  wearing  parts  accessible  at  all  times,  so  that  they  can  be  cleaned  or  replaced  by  new  parts 
if  necessary. 

Fig.  382  shows  the  elevation  and  plan  for  a  centre  bearing,  or  pivot.  When  the  pin  turns, 
the  sliding  takes  place  between  friction  disks,  which  are  usually  made  of  hard  steel,  phosphor- 


Fig.  381. 


*  This  is  about  the  limiting  length  for  shipping  a  girder  on  cars  with  safety.  Plate  girders  136  feet  long  have  been 
shipped  on  cars  for  a  short  distance,  but  it  is  dangerous  in  any  case,  and  especially  when  shipments  are  made  to  a  great 
distance  over  all  kinds  of  roads. 


358 


MODERN  FRAMED  STRUCTURES. 


Friction  Discs 


Center  Casting 


bronze,  or  gun-metal.    In  order  to  insure  that  the  sliding  shall  take  place  between  the 

disks  only,  the  upper  and  lower  disks  are  provided 
with  short  dowels  which  fit  in  corresponding  sockets 
in  the  pin  and  centre  casting,  and  prevent  their 
sliding.  The  surfaces  of  the  disks  are  grooved  where 
they  come  in  contact  with  each  other.  This  is  to 
insure  lubrication ;  the  recess  in  the  centre  casting 
being  kept  full  of  oil  at  all  times,  the  oil  finds  its  way 
into  the  grooves. 

The  safe  load,  on  the  disks,  may  be  taken  at  3000 
lbs.  per  square  inch,  when  the  bridge  is  turning.  On 
small  disks  the  pressure  may  be  as  high  as  6000  lbs. 
per  inch.  In  general,  the  safe  load  per  square  inch  on 
a  disk  depends  on  the  frequency  of  the  turning,  the 
angular  velocity,  and  the  lubrication  of  the  disk.  If 
the  disk  be  small,  so  that  the  oil  grooves  cut  a  com- 
paratively large  surface  from  the  disk  and  insure  lubri- 
cation, the  safe  load  may  be  higher  than  for  a  larger 
disk.  At  10,000  lbs.  pressure  per  square  inch  there  is 
danger  from  abrasion.  The  pin  and  centre  casting 
may  be  of  cast-iron  or  cast-steel. 

-When  a  riveted  truss  design  is  used  instead 
of  the  plate  girder  bridge,  the  centre  bearing  is  generally  the  same  as  for  a  plate  girder.  The 
great  advantage  in  using  the  riveted  truss  design  for  lengths  from  100  to  160  feet  is  that  the 
trusses  can  be  made  in  two  halves  complete,  and  then  coupled  up,  over  the  centre,  by  means 
of  eyebars  with  pin  connections. 

Fig.  383  illustrates  a  method  of  coupling  up  the  two  halves  of  a  riveted  truss  swing  bridge. 
The  upper  chord  AA  being  at  all  times  under  tension,  we  can  here  use  eyebars  in  the  place 


Fig.  382. 

370.  Riveted  Pony  Truss  Swing  Bridges. 


1  r 


ji 


Fig.  383. 


of  riveted  members.  The  lower  chord  at  B  being  at  all  times  under  compression,  a  butt  joint 
can  be  used,  with  such  splicing  as  is  ordinarily  used  for  such  joints.  The  great  saving  in  this 
design  is  in  the  facility  with  which  the  coupling  at  A  can  be  made  in  the  field.  The  wheels 
IV  are  balance  wheels,  usually  four  in  number,  to  balance  the  bridge  when  swinging.  These 
wheels  are  not  supposed  to  carry  any  direct  load.  They  are  simply  intended  to  carry  any 
unbalanced  load  when  the  bridge  is  swinging,  caused  by  wind,  or  the  unstable  equilibrium,  of 
the  bridge  itself.    The  usual  method  for  designing  these  wheels  is  to  assume  an  unbal- 


THE  DESIGN  OF  SWING  BRIDGES. 


359 


Fig.  384. 


anced  load  of  say  1000  lbs.  at  one  end  of  the  bridge  when  swinging.    As  their  duty,  at  all 
times,  is   not   severe,  they  are   usually  made   to  run 
on  a  heavy  T  rail,  which  is  bent  to  a  circle  and  laid 
directly  on  the  masonry. 

These  balance  wheels  are  necessary  in  all  forms  of 
centre-bearing  bridges,  whether  girder  or  truss  designs. 
All  of  the  previous  sketches  are  intended  for  through 
bridges.  When  the  distance  from  the  base  of  rail  to  the 
under  side  of  the  bridge  is  sufficient,  a  deck  bridge  is 
used.  The  method  of  supporting  a  deck  bridge,  at  the 
centre,  does  not  differ  materially  from  that  of  a  through 
bridge.  In  a  deck  bridge,  however,  it  is  desirable  to  have 
the  support  for  the  centre  as  high  up  as  is  possible,  that  is, 
as  near  to  the  centre  of  gravity  of  the  load  as  will  permit. 
This  necessitates  raising  the  centre  casting  up  to  the  required  level.  This  is  sometimes 
accomplished  by  placing  it  upon  a  pedestal.    The  usual  way,  however,  is  to  change  the  form 

of  the  centre  casting.  Fig.  384  shows  the  cross-section  of  a 
deck  plate  girder  swing  bridge  near  the  centre.  The  centre 
casting  has  now  the  shape  of  a  frustum  of  a  cone,  and  is 
called  the  cone. 

371.  Centre  Bearing  on  Conical  Rollers. — In  all  of 
J  the  previous  designs  we  have  contemplated  a  sliding  friction 
at  the  centre  by  means  of  friction  disks.  In  spans  where 
rapid  and  frequent  swinging  is  required,  it  is  desirable  to  get 
rid  of  the  sliding  friction  at  the  centre.  Here  we  put  the 
load  on  a  nest  of  conical  rollers.  Fig.  385  shows  a  section  in 
elevation  and  a  plan  of  a  centre  bearing  on  conical  rollers.* 
The  conical  rollers  R  are  made  of  hard  steel.  The  box  is 
made  of  cast-iron  or  cast-steel.  In  using  this  form  of  centre 
great  care  should  be  taken  to  have  the  bearing  on  the  rollers 
at  all  times  true  and  level.  Any  inequality  of  bearing  on 
moving  parts  results  in  unequal  wearing.  If  some  of  the 
rollers  become  worn  more  than  others,  owing  to  unequal 
bearing,  imperfect  workmanship,  or  other  causes,  they  soon 
lose  their  proper  relative  position  in  the  nest,  and  begin 
Fig.  385.  crowding  and  wearing  each  other.    Then  the  nest  becomes 

clogged  with  the  grindings,  and  the  efficiency  of  the  bearing  is  soon  destroyed. 

In  the  recent  designs  for  conical  rollers  this  trouble  is  anticipated,  and  in  order  to  guard 
against  it  the  rollers  are  encircled  by  a  collar,  or  live  ring2.?>  it  is  called.  Each  roller  is  then 
held  in  its  proper  position  by  a  small  rod  which  passes  through  its  centre.  The  outer  ends 
of  these  rods  are  fastened  to  the  live  ring.  The  inner  ends  come  together  on  a  hub  called 
the  spider.  This  device  insures  the  same  movement  in  all  of  the  rollers  at  the  same  time. 
The  safe  load  on  friction  rollers  is  given  by  the  formula  /  =  600  Art.  255.  For 
hard-steel  rollers  on  cast-iron  beds  use  /»  =  500'^^^-  For  conical  rollers  use  the  average 
diameter  for  d. 

The  bearing  on  the  masonry  at  the  centre  may  be  taken  at  the  same  values  as  for  fixed 
spans,  when  the  bridge  is  closed  and  fully  loaded  with  live  load.  When  the  bridge  is  empty 
and  swinging,  the  above  values  may  be  doubled.    For  example,  if  the  permissible  bearing  on 


This  form  of  centre  bearing  was  invented  by  Mr.  Wm.  Sellers,  and  is  known  as  the  Sellers  Centre. 


360 


MODERN  FRAMED  STRUCTURES. 


the  masonry  is  300  lbs.  per  square  inch  for  fixed  spans,  then  for  swing  bridges  it  would  apply 
only  when  the  bridge  is  closed  and  fully  loaded.  When  the  bridge  is  empty  and  swinging, 
we  may  use  double  this  value,  or  600  lbs.  per  square  inch.  The  practice  simply  demands  a 
higher  value  for  the  bearing  on  the  masonry  when  the  bridge  is  swinging  than  when  it  is 
closed,  and  about  double  the  value  seems  rational. 

372.  Truss  Swing  Bridges. — As  truss  spans  can  be  built  either  with  pin  or  riveted 
connections,  a  discussion  of  one  will  apply  to  both.  In  this  country  the  common  practice  is 
to  use  pin  connections  for  long  spans.  This  applies  to  swing  bridges  for  all  lengths  above 
those  which  are  too  long  for  using  the  plate  girder  or  riveted  truss  design.  In  what  follows, 
all  truss  bridges  will  be  understood  to  be  pin  connected.  All  trusses  for  swing  bridges 
should  be  simple  in  design,  so  as  to  be  free  from  all  ambiguities  in  their  stresses  under  all 
possible  conditions  of  loading  and  temperature,  as  well  as  the  conditions  resulting  from  a 
rapid  variation  in  the  depth  of  the  trusses,  which  necessarily  involves  the  moment  of  inertia 
of  the  cross-section.  In  all  of  the  analysis  on  swing  bridges  it  has  been  assumed  that  the 
moment  of  inertia  of  the  continuous  girder  is  constant.  Any  great  departure  from  this 
assumption  invalidates  the  correctness  of  the  assumption  and  vitiates  the  values  of  the  com- 
puted reactions.*  No  simple  classification  of  different  types  of  truss  swing  bridges  can  be 
made.  In  general,  the  most  preferable  design,  everything  else  being  equal,  should  be  a  single 
truss  system  free  from  all  adjustable  members.  The  advantage  of  using  long  panels  can  be 
obtained,  as  in  all  fixed  spans,  by  secondary  trussing.  Inclined  upper  chords  for  through 
bridges  are  introduced  where  economy  of  design  dictates  their  use.  In  general,  it  may  be 
said  that  the  outline  for  the  trusses  of  a  swing  bridge  is  governed  by  the  same  rules  as  for 
fixed  spans.  The  most  economical  depth  at  the  centre  is  about  the  same  as  for  a  fixed  span 
having  a  length  equal  to  the  total  length  of  the  swing  bridge. 

373.  Unusual  Forms  of  Trusses.- — Perhaps  a  good  way  to  illustrate  the  best  forms  for 
trusses  to  be  used  would  be  to  show  some  which  are  objectionable.    An  unusual  form  is  the 


Fig.  386. 


triangular  pattern.  In  this,  all  of  the  lines  of  the  truss  form  triangles.  Fig.  386  shows  the 
outline  for  a  triangular  form  of  truss. 

The  principal  objection  to  this  form  of  truss  is  the  variation  in  the  depth  of  the  truss 
from  zero  at  the  ends  to  the  maximum  at  the  centre.  The  low  depth  of  the  truss  at  the  ends 
not  only  vitiates  the  value  of  the  reactions  obtained  by  assuming  a  uniform  moment  of  inertia, 
but  it  also  results  in  excessive  deflections  when  the  bridge  is  closed  and  the  live  load  is  on  one 
arm.  These  excessive  deflections  tend  to  bend  the  chords.  As  the  chord  joints  are  not  artic- 
ulated, the  bending  may  be  suflficient  to  give  the  chords  a  permanent  set.  This  has  actually 
occurred  in  two  instances  where  this  form  of  truss  has  been  used.  However,  this  form  of 
truss  presents  some  advantages  which  should  not  be  lost  sight  of.  Tlie  low  inclined  chords 
at  the  ends  will  aid  in  deflecting  a  possible  derailed  car  which  might  strike  the  bridge.  The 
form  of  the  truss,  also,  is  such  as  to  reduce  the  areas  exposed  to  wind  pressure  at  the  ends  of 

*  For  the  ordinary  American  practice  the  errors  introduced  by  the  varying  moment  of  inertia  practically  compen- 
sate for  the  errors  coming  from  neglecting  the  web  members  in  computing  the  continuous  girder  moments.  See 
Art.  178a,  p.  196. 


THE  DESIGN  OF  SWING  BRIDGES.  361 

the  arms,  making  it  somewhat  easier  to  handle  during  high  winds.    This  form  of  swing 
bridge  was  at  one  time  very  commonly  used  for  short  spans,  but  has  now  given  way  to  more 
economical  forms,  more  especially  the  plate  girder  and  the  riveted  truss  design. 
Fig.  387,  shows  another  form  of  truss  for  swing  bridges  which  is  objectionable. 


Fig.  387. 


The  objection  to  this  form  of  truss  design  lies  principally  in  the  central  portion  directly 
over  the  turntable.  Here  the  character  of  the  design  involves  an  uncertain  distribution  of 
loading  on  the  turntable.  Again,  the  counter-stresses  in  the  web  system  are  provided  for  by 
the  use  of  adjustable  rods,  which  are  objectionable  in  any  truss  design,  owing  to  their  ten- 
dency to  get  out  of  adjustment.  However,  the  most  important  objection  to  this  form  of  con- 
struction is  the  lack  of  economy  in  the  design. 

374.  Standard  Forms  of  Trusses. — As  previously  stated  the  most  desirable  form  of  truss 
design  is  that  which  is  free  from  adjustable  members  and  from  all  ambiguities  in  the  stresses 
arising  from  complex  systems  of  trusses.    Fig.  388  shows  a  standard  form  of  truss  for  spans 


Fig.  388. 

of  any  length.  The  principal  advantages  in  this  form  are,  economy  in  design,  a  minimum 
number  of  members,  the  absence  of  all  adjustable  members,  a  freedom  from  all  ambiguities  in 
the  stresses,  etc.  In  this  design,  the  members  AD  should  have  such  an  inclination  that  when 
the  bridge  is  fully  loaded,  the  members  ^Z?  will  carry  the  greater  part  of  the  load  to  the  points 
B  on  the  turntable.  This  will  insure  a  better  distribution  of  the  maximum  load  on  the  turn- 
table. Another  advantage  in  this  form  of  truss  lies  in  the  use  of  the  links  shown  in  the  figure 
by  the  memberyj-^.  Aside  from  their  primary  use  in  equalizing  the  loads  on  the  turntable  at 
points  B,  they  have  proven  useful  in  case  the  bridge,  when  swinging,  becomes  unbalanced. 
Such  a  condition  may  arise  in  case  the  bridge  is  struck  by  a  passing  steamer,  which  may  lift 
one  arm,  while  the  other  arm  deflects  downward.  If  the  joints  at  B  will  yield  so  as  to  allow 
the  arms  to  deflect  considerably,  the  bridge  may  remain  intact  on  the  turntable.* 

There  are  a  great  many  standard  forms  of  trusses  for  swing  bridges ;  in  fact,  too  many  to 
be  described  here.  The  authors  have  shown  in  Fig.  388  what  they  consider  the  best  form  of 
truss  for  swing  bridges.  However,  conditions  may  arise  which  may  compel  the  designer  to 
modify  this  design.  For  instance,  in  single  track  swing  bridges  it  frequently  happens,  when  it 
becomes  necessary  to  handle  the  bridge  by  steam  power,  that  the  engine  must  be  placed  over 


*  Such  an  accident  occurred  to  the  swing  bridge  of  the  St.  Louis  Southwestern  Railway  over  the  Red  River  at 
Garland  City.    The  bridge  was  built  from  C.  Shaler  Smith's  plans. 


362 


MODERN  FRAMED  STRUCTURES. 


the  track.  In  that  event  it  becomes  convenient  to  have  vertical  centre  posts  to  support  the 
shaft  leading  vertically  down  from  the  engine-room  to  the  drum.  This  shaft  (there  are  some- 
times more  than  one  shaft)  is  always  vertical,  and  the  vertical  posts  at  the  centre  make  a  con- 
venient support  for  journal-boxes  holding  the  shaft.     This  condition  may  modify  the  form  of 


A  A 


V////////////M//////M 
Fig.  389. 


truss  to  that  shown  in  Fig.  389.  Here  the  centre  posts  AB  are  vertical,  and  the  engine-house 
is  usually  supported  on  the  cross-girder  CC. 

375.  Methods  of  Supporting  Truss  Swing  Bridges  at  Centre.— Truss  swing  bridges 
are  sometimes  made  centre  bearing,  similar  to  plate  girder  bridges,  where  the  available  depth 
from  base  of  rail  to  the  top  of  centre  pier  is  limited,  so  as  to  preclude  the  use  of  any  other 
form  of  support  at  the  centre.  However,  such  conditions  have  to  be  provided  for  by  special 
designs.  The  usual  method  of  supporting  the  trusses  of  a  swing  bridge  is  on  a  turntable. 
As  previously  stated,  if  the  trusses  of  a  swing  bridge  rest  on  a  circular  girder,  called  a  drum, 
which  in  turning  moves  upon  rollers,  the  bridge  is  said  to  have  a  rim-bearing  turntable.  The 
word  Turntable  comprises  the  entire  turning  arrangement  at  the  centre.    Fig.  390,  shows  the 


— \ 

f  1 

•\  f\t\  r\  i  \  r 

 f  

■\r\  n  r\  i\  nrii\M 

iroOQOOO  O  OQOOOOOOOIII, 


Fig.  390. 


Fig.  391. 


outline,  in  plan  and  elevation,  of  a  rim-bearing  turntable  loaded  at  four  points  marked  P. 
This  form  of  loading  on  a  turntable  is  now  avoided  excepting  perhaps  for  highway  bridges 

The  objections  to  loading  a  bearing  rim  at  four  points  only  is  that  it  is  impossible  to  get 
3  uniform  load  on  the  rollers,  no  matter  how  stiff  we  make  the  circular  girder  or  drum. 


THE  DESIGN  OF  SWING  BRIDGES. 


363 


The  fact  is,  in  this  form  of  turntable,  the  wheels  which  are  directly  under  the  loaded 
points  carry  the  entire  load.  The  result  is  that  the  rollers  and  the  tread  upon  which  they  roll 
soon  show  signs  of  unequal  wearing. 

The  common  method  of  loading  a  rim-bearing  turntable  is  to  distribute  the  load  from  the 
trusses  equally  over  eight  symmetrical  points.  Fig.  391,  shows  the  outline,  in  plan  and 
elevation,  for  a  rim-bearing  turntable  loaded  at  eight  points.  The  loads  from  the  trusses 
come  on  the  four  corners  of  a  square  box  PPPP.  The  four  sides  of  this  box  are  the  girders 
which  rest  directly  on  the  drum  at  the  eight  points  marked  B.  If  the  loads  at  P  are  equal, 
the  loads  at  B  must  be  each  equal  to  ^P.  The  points  B  in  the  drum  can  be  spaced  at  equal 
distances,  by  assuming  the  proper  diameter  for  the  drum.  If  a  equals  the  side  of  the  square 
PPPP,  then  the  diameter  of  the  circle,  necessary  to  space  the  point  B  at  equal  distances  on 
the  drum,  =  1.0824a. 

Plate  IV  shows  the  general  design  for  a  rim-bearing  turntable. 

All  swing  bridges,  whether  centre  bearing  or  rim  bearing,  must  rotate  about  some  fixed 
point  or  centre  pin.  In  a  purely  rim-bearing  turntable  this  centre  pin  carries  no  load.  Its 
duty  is  simply  to  hold  the  turntable  in  position  with  reference  to  its  centre  of  rotation.  The 
fact  that  the  centre  pin  carries  no  load  is  one  of  the  objections  to  the  rim-bearing  turntable. 
The  centre  pin  usually  turns  in  a  centre  casting  which  rests  directly  on  the  masonry  (see 
Plate  IVa.  If  the  rollers  all  travelled  in  the  proper  circle,  the  centre  pin  would  not  tend  to 
move  laterally.  As  a  matter  of  fact,  the  rollers  are  constantly  getting  out  of  line  or  out 
of  their  proper  circle,  so  that  the  centre  pin  is  frequently  subjected  to  a  series  of  displacements 
laterally.  If  the  loads  are  comparatively  light,  the  pin  and  centre  casting  can  be  made  to 
withstand  this  tendency  to  displacement  by  anchoring  the  centre  casting  to  the  masonry. 

The  most  approved  design  for  a  turntable  has  the  weight  distributed,  so  that  a  portion  of 
the  loads  goes  to  the  centre  pin.  This  form  of  turntable  is  called  the  centre-bearing  and  rim- 
bearing  turntable  combined. 

p  p 

Jf  L 

A  A  A  A 


Fig.  392. 

In  this  kind  of  a  turntable,  the  load  on  the  centre  pin  not  only  relieves  the  load  on  the 
rim,  whereby  the  bridge  turns  easier,  but  it  also  aids  in  holding  the  centre  pin  in  its  proper 


564 


MODERN  FRAMED  STRUCTURES. 


position,  and  in  this  way  helps  to  keep  the  rollers  in  the  proper  circle.  Fig.  392  shows  the 
outline,  in  plan,  and  elevation  for  a  rim-  and  a  centre-bearing  turntable.* 

The  loads  from  the  trusses  come  on  the  four  corners  of  the  square  box  PPPP.  The 
four  sides  of  this  box  are  the  girders,  which  rest  on  eight  points  marked  A.  The  four  corners 
of  the  box  are  perfectly  free;  so  that,  although  the  corners  /'come  directly  over  the  circle  of 
the  drum,  there  is  no  load  transmitted  to  the  drum  directly  under  the  point  P.  The  eight 
points  A  now  carry  the  entire  load  from  the  four  points  P.  The  load  at  A  is  then  transmitted 
to  the  two  adjacent  radial  girders  BC,  and  these  girders  in  turn  carry  a  portion  of  the  load  to 
the  centre,  and  the  remainder  to  the  rim.  In  this  manner,  the  load  which  comes  on  the  rim 
is  uniformly  distributed  over  sixteen  points.  This  form  of  turntable  gives  the  best  distribution 
of  loads,  and  consequently  turns  easier  than  any  other  form  of  turntable  in  use  in  this  country ; 
that  is,  excepting,  of  course,  the  purely  centre-bearing  swing  bridges,  which,  properly  speaking, 
have  no  turntable. 

The  safe  load  on  static  rollers  may  be  taken  the  same  as  given  for  small  rollers.  Chap. 
XVIII ;  that  is,/  =  1200  ^ d.  The  safe  load  on  moving  rollers  may  be  taken  at  /  =  600  ^ d. 
This  value  of  p  has  ample  margin  for  the  variation  in  the  hardness  of  the  different  materials 
used  for  rollers  and  treads,  unequal  bearing  of  rollers,  etc. 

376.  End  Lifting  Arrangements. — In  the  early  designs  for  swing  bridges  very  little 
attention  was  paid  to  the  proper  elevation  of  the  ends  of  the  arms  when  the  bridge  is  closed  ; 
and  even  at  this  late  day  a  great  many  important  swing  bridges  are  built  by  contractors  with 
an  utter  disregard  of  the  condition  of  the  end  supports.  In  fact,  the  majority  of  swing  bridges 
have  no  provisions  for  lifting  the  ends  whatever;  while  others  have  all  kinds  of  make-shifts, 
which  generally  shirk  their  duty  entirely.  It  is  safe  to  say  that  in  this  country  it  is  the  ex- 
ception to  find  a  swing  bridge  where  proper  provision  is  made  for  raising  the  ends  when  the 
bridge  is  closed.  It  has  been  shown,  in  Chap.  XII,  that  if  the  ends  are  not  raised  we  cannot 
obtain  the  conditions  necessary  for  a  beam  continuous  over  three  rigid  supports.  In  other  words, 
if  the  ends  are  not  raised,  we  must,  in  our  analysis  for  finding  the  stresses,  make  an  assump- 
tion which  will  satisfy  this  condition  of  the  ends  under  the  extreme  variations  of  temperature. f 
Now,  it  is  difificult  to  say  just  what  this  condition  should  be.  Furthermore,  if  the  ends  are  left 
free  to  hammer,  under  extreme  variations  of  temperature,  the  ends  may  be  thrown  out  of 
line  so  far  as  to  cause  derailment  of  a  train  coming  on  the  bridge.  In  fact  a  swing  bridge, 
wherein  no  proper  provision  is  made  for  raising  the  ends,  is  a  dangerous  structure  at  all 
times. 

In  the  previous  analysis  on  swing  bridges,  Chap.  XII,  two  conditions  of  the  ends  were 
assumed,  in  computing  the  stresses  due  to  dead  load  only.  First,  Case  I,  when  the  ends  were 
just  touching  their  supports  without  producing  any  positive  reactions.  Then,  in  Case  II,  the 
ends  were  assumed  to  be  raised,  so  that  the  reactions  were  equivalent  to  those  of  a  beam 
continuous  over  three  level  supports.  Now,  inasmuch  as  we  combine  the  live  load  stresses 
with  either  Case  I  or  Case  II,  it  follows  that  the  ends  should  at  all  times  be  raised  so  as  to 
satisfy  a  mean  between  those  conditions  assumed  for  Cases  I  and  II.  The  ends  could  probably 
be  raised  high  enough  to  satisfy  Case  II  for  the  time  being,  but  under  extreme  variations  of 
the  relative  temperature  of  the  two  chords  this  condition  would  be  changed  ;  so  that,  in  order 
to  satisfy  all  conditions  of  temperature  as  well  as  loading,  the  ends  must  be  raised  enough  to 
take  out  at  least  one  half  the  deflection  of  the  ends  of  the  arms  due  to  dead  load  under  Case 


*  This  form  of  turntable  is  now  used  exclusively  by  the  Detroit  Bridge  and  Iron  Works. 

f  By  "  variations  of  temperature  "  in  this  chapter  is  meant  simultaneous  differences  of  temperature  of  the  two 
chords  (upper  and  lower)  of  a  drawbridge.  The  lower  chord  may  be  entirely  in  the  shade  while  the  upper  chord  is  en- 
tirely in  the  sun,  and  this  may  cause  a  difference  of  temperature  in  the  two  chords  of  40°  or  50°  F.  At  night  the  vari 
s.tion  may  be  in  the  opposite  direction. 


THE  DESIGN  OF  SWING  BRIDGES. 


365 


I,  span  swinging.  This  would  still  leave  the  ends  of  the  arms  below  the  true  level  of  the 
lower  chord  ;  that  is  to  say,  one  half  the  deflection  would  still  remain  in  the  truss. 

The  proper  way  to  eliminate  this  remaining  deflection  in  the  arms  is  to  shorten  the  upper 
chord,  usually  in  the  centre  panel  or  those  adjacent  to  the  centre,  just  enough  to  raise  the 
ends  to  the  true  level  of  the  lower  chord. 

For  example,  assume  a  swing  bridge  400  feet  long  and  50  feet  deep  over  the  centre 
The  computed  deflection  of  the  ends  of  the  arms  under  Case  I  is,  say,  4  inches.  Now  if  the 
ends  were  raised  2\  inches  by  means  of  the  end  lifts,  there  remains  if  inches  of  the  end  de- 
flection to  be  eliminated  by  shortening  the  upper  chord.  The  amount  that  the  upper  chord 
must  be  shortened  on  each  side  of  the  centre  is  if  X  —  c>-44  inch  =  y\  of  an  inch.  This 
subject  will  be  taken  up  in  detail  further  on. 

The  simplest  form  of  an  end  lift  for  a  swing  bridge  is  that  in  which  the  ends  of  the  arms 
are  provided  with  wheels  which  rest  on  the  crowns  of  inclined  roller  beds.  Fig.  393  is  an 
elevation  of  the  end  of  a  truss  swing  bridge,  showing  the  wheels  R  resting  on  the  beds  B, 
which  are  bolted  to  the  masonry. 


Fig.  393. 

The  roller  beds  R  are  set  at  a  proper  elevation  to  give  the  end  reactions,  as  previously 
shown.  In  this  form  of  an  end  lift  it  becomes  necessary  to  use  some  form  of  automatic  latch, 
to  stop  the  wheels  from  rolling  over  the  top  of  the  beds  and  down  on  the  other  side.  When 
the  span  is  long,  so  that  it  can  acquire  considerable  momentum  in  turning,  the  automatic 
latch  can  no  longer  be  relied  on  to  bring  the  bridge  to  a  sudden  stop.  The  fact  that  the  ends 
must  be  lifted  by  the  very  energy  which  the  bridge  acquires  in  turning  implies  that  the 
bridge  must  have  a  good  angular  velocity  when  near  closing,  so  as  to  force  the  wheels  up  the 
inclined  beds.  This  energy,  or  shock,  which  is  taken  up  by  the  latch  is  transmitted  to  the 
masonry.  For  short  spans,  where  the  ends  are  to  be  lifted  about  \\  inches  or  less,  this 
method  of  lifting  is  effective  and  inexpensive.  It  is  especially  adapted  for  use  in  short  spans, 
which  have  to  be  opened  quickly;  the  rollers  being  once  started  down  their  inclined  beds,  the 
same  energy  is  imparted  to  the  moving  mass  of  the  bridge  which  was  necessary  to  raise  the 
ends. 

When  it  becomes  necessary  to  lift  the  ends  of  a  plate-girder  swing  bridge,  it  is  customary 
to  fix  the  end  wheels  on  the  masonry,  and  provide  the  ends  of  the  arms  with  beveled  plates, 
to  correspond  with  the  inclined  beds  previously  shown.    Fig.  394  gives  in  elevation  the  end 


Fig.  394. 

of  a  plate-girder  swing  bridge,  showing  three  stands  of  end  wheels,  with  two  wheels  in  each 
stand.    The  inside  stand  of  wheels  carries  no  load  of  any  kind  when  the  bridge  is  closed. 


366 


MODERN  FRAMED  STRUCTURES. 


Its  duty  is  merely  to  assist  in  steadying  tlie  bridge  when  coming  on  or  rolling  off  the  end 
wheels.  It  should  be  stated  here  that  for  all  short-span  swing  bridges  the  ends  should  be 
raised  relatively  higher  than  for  long  spans.  For  example,  if  the  theoretical  deflection  of  the 
ends  due  to  dead  load,  Case  I,  were  such  that  we  should  have  to  give  the  ends  only  a  ^-inch 
lift,  for  practical  reasons  we  would  make  it  probably  50  per  cent  more,  or  f  inch,  to  allow 
for  extreme  variations  in  temperature,  which  cannot  be  provided  for  by  any  assumptions  that 
we  might  make. 

There  are  other  simple  forms  of  end  lift,  such  as  the  wedge,  for  instance,  which,  however, 
do  not  lift  the  ends,  and  therefore  do  not  properly  belong  here.  What  they  do  is  simply  to 
give  the  ends  of  the  bridge  a  firm  bearing  for  certain  conditions  of  temperature,  and  prevent 
the  ends  from  hammering.*  Under  certain  conditions  of  temperature  changes,  the  ends 
may  not  touch  their  supports  at  all,  which  necessitates  a  new  adjustment  for  the  wedges. 
Again,  under  other  conditions  of  temperature,  the  wedges  may  be  held  so  firmly  in  place  as 
to  make  it  difficult  to  draw  them  out  from  under  the  ends  of  the  arms,  so  that  any  form  of 
end  lift  which  has  to  be  adjusted  for  different  conditions  of  temperature  is  usually  left  out  of 
adjustment,  because  it  is  impracticable  to  handle  the  bridge  otherwise. 

There  is  another  kind  of  end  lift,  suitable  for  short  spans,  which  should  be  explained  here. 
In  the  end  lift  just  described  the  shock  transmitted  to  the  masonry,  when  closing  the  bridge, 
may  be  so  objectionable  as  to  preclude  its  use.  The  form  shown  in'  Fig.  395  should  then  be 
used. 


Fig.  395. 

In  this  form  of  end  lift  the  bridge  is  first  swung  into  line  and  latched  ;  then,  by  turning 
the  vertical  shaft  A,  the  wheels  B  revolve  about  their  centres,  and  at  the  same  time  force  the 
suspended  wheels  W  under  the  ends  of  the  arms. 

Let  Fig.  396  represent  the  motion  of  the  parts  of  the  end  lift  shown  in  Fig.  395.  Let  li 
represent  the  required  lift  of  the  arms  in  feet.    Let  R  represent  the  load  on  wheel  W  when 


Fig.  396. 


R 


the  maximum  lift  has  been  attained.    Then  the  total  work  done  in  raising  the  end  =  ~  h. 

2 

As  the  time  for  opening  or  closing  a  swing  bridge  is  usually  limited,  let  t  equal  the 
numb.er  of  seconds  required  to  raise  the  ends ;  then  the  mean  power  required  to  raise  the  ends 

^RXh 

2t 


*  See  Plate  IV. 


THE  DESIGN  OE  SWING  BRIDGES. 


367 


This  form  of  end  lift  is  well  adapted  for  use  in  spans  under  250  feet  long,  where  the 
distance  at  the  abutment  from  the  base  of  the  rail  to  the  masonry  is  limited. 

The  Screw-jack  or  Direct  Lift. — In  this  form  of  end  lift  the  load  at  the  ends  is  lifted 
directly  by  means  of  screw-jacks  placed  at  the  four  corners  of  the  bridge.  These  jacks  are  usually 
four  in  number.  They  are  ail  connected  by  means  of  shafting  and  gearing,  and  are  worked  by 
a  motor  from  the  centre  of  the  bridge.  The  screw-jack  is  well  adapted  for  use  in  long  spans 
where  the  value  of  h,  or  the  distance  through  which  the  ends  must  be  lifted,  is  a  large  factor. 

R  y,  h 

The  mean  power  required  to  raise  the  ends  is,  as  before,  — .    Fig.  397  shows  a  form  of 

screw-jack  under  one  end  of  a  bridge. 

No  attempt  is  here  made  to  show  the  actual  form  of  screw-jack  as  generally  used,  but 
the  illustration  is  adapted  to  show  simply  how  a  screw-jack  could  be  made  to  exemplify  the 
principles  used.  In  Fig.  397,  if  the  wheel  Wis  turned  we  turn  the  screw  in  or  out  of  the 
cylinder  C,  which  correspondingly  lowers  and  raises  the  end  of  the  span.  The  lower  end  of 
the  screw-shaft  turns  in  a  socket  in  the  pedestal  P. 

In  any  form  of  lift  wherein  the  multiple  of  force  of  the  moving  parts  of  the  mechanism 
remains  a  constant,  the  power  required  varies  directly  with  the  load.  If  the  load  is  uniformly 
varying,  the  power  increases  uniformly  to  the  end  of  the  lift. 


'      s  /  ' 


iniiiiiiiiii 

III  mill 

w 


Fig.  397. 


Fig.  398. 


In  Fig.  398,  let  /  be  the  half  length  of  a  swing  bridge  and  zvl  =  W  the  weight  of  one  arm. 
Let  R  represent  the  final  force  applied  to  the  end  lift,  which  we  can  assume  to  be  a  screw- 
jack.    Then  the  deflection 


wr 


Rr 


in  which  d  is  positive  downward. 
From  eq.  (i)  we  have 


R  = 


W-dx 


(2) 


In  eq.  (2),  when  d  =  o,  R  —  ^W \  which  is  the  reaction  for  a  beam  continuous  over  three 
supports  and  uniformly  loaded. 

It  has  been  shown  that  the  full  amount  of  the  end  deflection  should  not  be  taken  out  by 
the  end  lifts,  but  rather  only  one  half  this  amount,  or  perhaps  a  little  more  than  one  half,  to 
allow  for  variations  of  temperature  of  the  two  chords;  so  that./?  should  not  be  f fF,  but 


368  MODERN  FRAMED  STRUCTURES. 

rather  nearer  -^^W.  The  value  of  R  then  varies  from  zero  to  about  -i§W;  and  in  any  form  of 
end  hft,  such  as  the  screw,  the  force  applied  would  vary  in  the  same  ratio. 

Now,  in  order  to  equalize  the  horse-power  expended  in  lifting  the  ends,  the  speed  of  the 
engine  should  vary  inversely  as  the  force  required.  This  means  that  the  engine  would  some- 
times be  racing,  while  near  the  end  of  the  lift,  when  R  becomes  nearly  IV,  tiie  piston  speed 
would  be  very  much  reduced  ;  while  under  unfavorable  conditions  the  engine  might  become 
stalled.    To  preclude  the  possibility  of  such  an  occurrence,  the  engine  should  be  able  to 

develop  the  full  power  required  at  the  average  piston  speed  :  that  is,  — y--'  =  mean  power 

required  ;  wherein  R  =  j^lV,  /.=  time  for  lifting  in  seconds,  and  /i  is  the  value  of  d*  in  equa- 
tion (2)  when  R  = -^^W.  Now,  it  frequently  so  happens  that  the  maximum  horse-power 
required  is  for  lifting  the  ends,  and  not  for  turning  the  bridge ;  so  that  probably  an  engine  of 
less  capacity  could  be  used  if  some  means  were  employed  by  which  the  multiple  of  force  used 
in  the  end-lifting  arrangement  were  made  to  vary  with  the  load.  This  is  practically  accom- 
plished in  the  end  lift  known  as  the  ram,  which  is  described  further  on. 

The  objections  to  the  screw-jack  apply  in  general  to  all  similar  forms  of  end  lifts — that  is, 
where  no  provision  is  made  for  varying  the  multiple  of  force  with  the  load.  Notable  among 
the  similar  forms  of  end  lifts  is  the  hydraulic  jack.  Here  the  load  is  lifted  by  a  hydraulic 
ram  instead  of  a  screw ;  and  the  medium  for  transmitting  power  is  glycerine  or  alcohol, 
instead  of  gearing  and  shafting.  Although  this  is  a  most  useful  means  of  doing  work  in  all 
forms  of  mechanism,  it  is  objectionable  for  lifts  for  swing  bridges  on  account  of  its  tendency 
to  leak.  Swing  bridges  have  been  built  which  can  be  raised  and  lowered  bodily  by  means  of 
hydraulic  jacks  at  the  centre.  The  work  done  in  raising  a  bridge  bodily  is  considerably  more 
than  the  work  done  in  raising  the  ends.  Their  ratio  is  about  as  five  to  one.  Among  such 
examples  may  be  mentioned  the  Harlem  River  Bridge  of  the  Manhattan  Elevated  Railway  in 
New  York.f    This  bridge  is  lifted  bodily  from  the  centre  by  means  of  a  hydraulic  jack. 


1 1 

r 
L. 

r 
1 

F^-  

1  ] 

[  [ 

0 

\ 

J- 

N 

< 

W///}/////M^//M/////////M.  W///)////m//mm/mLm 

Fig.  399. 

377.  The  Ram  with  Toggle-joints. — In  this  form  of  end  lift  the  multiple  of  force  is 
varied,  so  that  it  is  a  minimum  at  the  beginning  of  the  lift  and  a  maximum  at  the  end. 

Fig.  399  shows  in  outline  the  form  of  mechanism  employed  in  the  ram.    No  attempt  is 

*  This  value  of  d,  however,  is  obtained  in  another  way  more  accurately  by  computing  the  actual  deflection  of  the 

=  d 

ends  of  the  arms,  and  taking  h  -. 

>  2 

•f  This  bridge  was  designed  by  Mr.  Theodore  Cooper,  C.E. 


THE  DESIGN  OF  SWING  BRIDGES. 


369 


here  made  to  show  the  actual  form  of  the  parts  used.  The  screws  S,  one  at  each  end  of  the 
bridge,  are  turned  by  means  of  shafting  and  gearing  from  the  centre  of  the  bridge.  As  the 
screws  turn,  the  nuts  N,  called  the  cam  nuts,  travel  in  guides,  up  or  down,  on  the  screws, 
which  action  shortens  or  lengthens  the  toggle-joints,  and  at  the  same  time  lowers  or  raises  the 
ends  of  the  bridge. 

In  Fig.  400,  let  R  represent  the  end  reaction,  which  equals  the  load  to  be  lifted.  As  pre 
viously  shown,  R  may  vary  from  zero  to  ^^W.  Let  a,  b,  c  represent  the  toggle-joint  fixed  at 
a.    The  toggle  is  worked  in  and  out  by  means  of  a  horizontal  bar  H,  which  is  coupled  at  e  to 


Fig.  400. 

the  arms  fe  and  ed.  The  distance  dd'  represents  the  travel  of  the  cam  nut,  which  receives  a 
thrust  =:  P  from  the  screw. 

Let  F  represent  the  thrust  in  the  arm  ed  from  the  cam  nut  at  d ;  then  we  have 


F  —  —  cosec  Q, 
2 


p 

S  =  -  sec  <p* 
2 

H  =  F  cos  ^  +  .S  sin  0; 


.♦.  H  =  -(cot  0  4-  tan  0). 


But 


//=  2/?  tan  0; 


.:P=4Ri  --^  )  

Vcot  H  +  tan  0/ 

From  eq.  (3)  we  see  that  when  R  —  o,  P=o;  also,  when  0  =  o  we  again  have  P  =  o. 
That  is  to  say,  P,  which  is  the  thrust  on  the  cam  nut,  and  consequently  may  represent  the 
effort  of  the  motor  in  lifting  the  ends,  is  zero  for  minimum  and  maximum  values  of  R. 

In  Fig.  401,  the  curve  Oad  represents  the  effort  of  the  engine  during  a  piston  travel  Od, 
when  the  lifting  arrangement  is  such  as  previously  described.  The  straight  line  Oe  represents 
the  effort  of  the  engine  during  the  same  piston-travel  when  lifting  the  ends  with  a  direct  lift, 
such  as  the  screw-jack. 

*  Since  the  vertical  component  in      =  vert.  comp.  in  eJ  —  — ,  as  ie  is  horizortal 

2 


370 


MODERN  FRAMED  STRUCTURES. 


From  the  figure  we  see  that  the  maximum  effort  of  the  engine  in  the  first  case  is  much 
less  than  in  the  second — which  proves  that  the  ram  is  a  better  form  of  Hft  than  the  screw. 

The  dotted  curve  Oa'd'  represents  the  curve  of  effort  for  an  end  lift,  wherein  the  horizon- 
tal bar  H  described  in  Fig.  400  is  omitted,  and  where  deb,  which  is  now  one  straight  member, 
comes  into  a  horizontal  position  when  abc  becomes  vertical.    This  curve  shows  that  without 


Fig.  401. 

the  horizontal  bar  H  the  ram  is  not  as  efficient  as  the  screw,  for  which  the  curve  of  efYort  is 
represented  by  the  straight  line  Oe' . 

These  curves  were  all  plotted  from  actual  examples,  and  clearly  show  the  advantage  in 
using  the  ram  as  compared  with  any  other  known  form  of  end  lift.  For  any  given  case  the 
areas  of  the  figures  Oad  and  Oa'd',  and  the  triangles  Oed  and  Oe'd',  are  all  equal  to  each 

other,  as  they  represent  the  total  effective  work  done,  regardless  of  the  manner  in 

which  the  ends  are  lifted. 

378.  Machinery  for  Operating  Swing  Bridges. — The  machinery  for  operating  swing 
bridges  comprises  all  the  machinery  necessary  for  raising  and  lowering  the  ends,  as  well  as 
for  turning  the  bridge.  Swing  bridges  are  usually  operated  either  by  hand  power  or  steam 
power.  Electricity  has  been  applied  for  operating  them ;  but  here,  as  in  all  similar  applica- 
tions of  electricity,  it  is  only  a  means  of  transmitting  power  from  the  steam-engine  or  the 
turbine.  Gas-engines  have  also  been  employed  for  driving  the  machinery  for  swing  bridges; 
but  in  any  case,  whatever  may  be  the  source  of  the  power  employed,  the  motor  which  drives 
the  moving  parts  of  the  machinery  must  be  attached  to  the  motor  shaft,  so  that,  in  designing 
the  machinery  necessary  for  operating  a  swing  bridge,  the  kind  of  power  to  be  employed  need 
not  enter  into  consideration  at  all  until,  perhaps,  we  come  to  provide  the  necessary  space  for 
the  motor,  its  attachments  to  the  motor  shaft,  etc. 

The  simplest  means  for  turning  a  swing  bridge  would  be  to  pull  the  ends  of  the  bridge 
into  position,  for  opening  or  closing,  by  means  of  a  rope  attached  to  one  end  ;  and  this  is  a 
good  method  to  employ  for  turning  any  swing  bridge  whenever  the  machinery  becomes  dis- 
abled and  no  other  means  are  at  hand  for  applying  the  available  steam  or  hand  power. 

The  force  necessary  for  turning  is  usually  applied  to  the  circular  girder  or  drum  in  rim- 
bearing  bridges.    In  centre-bearing  bridges  the  force  is  applied  to  some  member  of  the  bridge 


THE  DESIGN  OE  SWING  BRIDGES. 


37* 


proper,  usually  a  cross-girder  or  floor-beam.  It  frequently  happens  that  one  arm  of  a  swing 
bridge  moves  over  dry  land.  In  that  case  it  may  be  convenient  to  apply  the  force  necessary 
for  turning  at  one  end  of  the  bridge.  This  will  enable  the  designer  to  place  all  of  the 
machinery  for  operating  the  bridge  on  dry  land.  However,  for  many  reasons  it  is  desirable 
to  have  the  motor  and  all  the  necessary  machinery  on  the  bridge  self-contained,  so  that  the 
foregoing  plan  cannot  be  recommended. 

In  this  article,  only  such  machinery  will  be  described  as  is  ordinarily  used  in  operating 
swing  bridges;  that  is,  where  the  force  necessary  for  turning  is  applied,  near  the  centre  of  the 
bridge,  through  one  or  more  pinions  which  mesh  with  the  teeth  on  a  circular  rack.  The 
pinions  are  keyed  to  vertical  shafts,  and  when  more  than  one  pinion  is  used  they  are  usually 
placed  in  pairs  diametrically  opposite  to  each  other.  The  vertical  shafts  are  connected 
through  bevel  gears  with  a  horizontal  shaft,  which  in  turn  is  connected  by  gearing  to  the 
motor  shaft,  usually  called  the  engine  shaft. 

379.  Resistances. — The  work  to  be  done  in  operating  a  swing  bridge  is :  first,  the 
lifting  of  the  ends,  which  of  course  includes  the  work  done  in  overcoming  the  resistances  of 
all  the  moving  parts,  as  well  as  the  internal  friction  of  the  engine  ;  second,  the  work  done  in 
turning  the  bridge,  which  includes  the  work  done  in  overcoming  the  resistances  due  to  inertia, 
wind  pressure,  and  the  resistances  of  all  the  moving  parts.  The  frictional  resistances  are: 
rolling  friction  between  the  rollers  and  the  top  and  bottom  treads ;  sliding  friction  between 
the  disks  under  the  centre  pin,  and  between  the  collars  on  the  spider-rods  and  the  rollers  ;  and 
the  resistances  of  the  machinery,  which  may  include  both  rolling  and  sliding  friction.  The 
resistance  of  inertia  is  the  force  required  to  accelerate  the  motion  of  the  bridge.  The  resist- 
ance due  to  wind  pressure  is  indeterminate  and  can  only  be  arrived  at  approximately.  If  the 
wind  blew  perfectly  steady,  so  that  the  pressure  is  uniformly  distributed  over  the  exposed 
area,  it  would  offer  no  resistance  to  the  turning  of  the  bridge,  as  the  pressure  on  one  side  of 
the  centre  would  equalize  that  on  the  other  whatever  the  direction  of  the  wind.  The  wind, 
however,  never  blows  steadily,  but  in  gusts,  and  the  pressure  is  never  uniformly  distributed, 
so  that  for  a  long-span  swing  bridge  the  velocity  at  one  end  seldom  equals  that  at  the  other. 
When  it  becomes  necessary  to  open  and  close  a  swing  bridge  in  all  kinds  of  weather,  tiie 
motor  used  should  be  capable  of  developing  sufficient  power  to  turn  the  bridge  when  the 
maximum  and  minimum  velocities  of  the  highest  wind  under  which  it  is  considered  safe  to 
operate  a  swing  bridge  are  acting  on  the  opposite  ends  at  the  same  time. 

Although  it  will  be  shown  that  the  unbalanced  wind  pressure  may  offer  the  greatest 
resistance  to  turning  and  frequently  determines  the  horse-power  to  be  used  in  operating  the 
bridge,  the  various  resistances  to  be  overcome  will  be  here  considered. 

In  what  follows  it  will  be  convenient  to  reduce  all  the  various  resistances  to  lifting  and 
turning  to  work  done  in  foot-pounds  per  second,  including  the  losses  due  to  the  transmission 
of  power  through  the  gearing,  etc. 

Lifting  the  ends. 

Let  R  —  -^^W  represent  the  load  to  be  lifted ; 
h  —  the  total  lift  in  feet ; 
t  —  time  in  seconds  for  lifting. 

Then,  since  the  maximum  rate  at  which  the  work  is  done  is  that  at  the  end  of  the  lift 
where  the  reaction  is  R,  we  have  for  the  power  required, 

_  RXh  zW/i 

rower  =   =  — ^ — . 

t  i6t 

Th'is  represents  the  maximum  rate  at  which  the  work  is  done,  in  foot-pounds  per  second, 
at  the  ends  which  are  to  be  lifted,  and  does  not  include  the  power  lost  in  the  resistances  of 
the  moving  parts  of  the  machinery. 


372 


MODERN  FRAMED  STRUCTURES. 


Turning  the  Bridge. — In  determining  the  power  necessary  for  turning,  it  will  be  most 
convenient  to  first  find  the  resistances  to  turning,  and  tlien  determine  their  equivalents  at  the 
pitch  circle  of  the  rack,  since  it  is  at  this  point  that  the  force  to  overcome  them  is  applied. 
Rolling  Friction. 

Let  R^  —  radius  of  the  drum  ; 
R  =  radius  of  rack  circle  ; 
Wr  —  weight  on  rollers  under  drum ; 
0,  =  coefificient  of  rolling  friction  ; 

Fr  =  force  at  rack  required  to  overcome  rolling  friction. 
Then 

K=cl>,Wr^  (4) 

Sliding  Friction  betweeti  Disks. 

Let  R  —  radius  of  the  rack  circle; 
—  weight  on  the  centre  pin; 
d  —  diameter  of  disks  ; 

0,  =  coefificient  of  sliding  friction  between  disks ; 

=  force  at  rack  required  to  overcome  sliding  friction. 
Then 

^.  =  0,^^.^  •  (5) 

Collar  Friction. — For  the  friction  of  the  washers  and  collars  at  the  ends  of  the  spider  rods, 


Let  r,  = 

interior  radius  of  collar; 

^  = 

exterior  radius  of  collar  ; 

r  — 

radius  of  the  rollers  ; 

coefificient  of  collar  friction. 

weight  on  the  rollers; 

R  and  R^  the  same  as  before ; 
F,  =  force  at  rack  to  overcome  collar  friction. 

2,T 

Then  the  force  with  which  the  rollers  are  pressed  against  the  collars  =  X 

From  mechanics,  the  lever  arm  of  the  friction  or  the  radius  of  the  circle  at  which  the 

2  r  '  — 

total  friction  may  be  considered  to  act  =  —  X   ^• 

3  ^  -  ^. 

The  ratio  of  the  force  at  the  rack  to  the  force  at  the  centre  of  the  track  required  to  over- 

come  the  collar  friction  =  -j^.  Then 

R 

r:  -  r:  ^2r^  R  ^  '^'^'■^  ^  R,  -  i  ^  r^  -r,'^    R  '  ' 

Wind. — To  find  the  effect  of  the  wind,  let  p  —  the  unbalanced  wind  pressure  acting  on 
one  arm,  and  let  /—  the  distance  of  the  centre  of  pressure  on  any  member  from  the  vertical 
plane  through  the  centre  of  rotation,  and  parallel  to  the  direction  of  the  wind,  the  direction  of 
the  wind  being  assumed  to  be  normal  to  the  plane  of  the  truss;  let  a  =  area  of  that  member, 
and  let  F^  —  a.  force  which  placed  at  the  rack  would  balance  the  total  unbalanced  wind 
pressure.    Then,  since  R  —  the  radius  of  the  rack  circle,  we  have 

2{pa/) 


THE  DESIGN  OF  SWING  BRIDGES. 


373 


However,  it  is  sufficiently  accurate  to  consider  the  total  wind  pressure  as  acting  at  a 
distance^  from  the  centre  —  one  quarter  the  length  of  the  bridge  ;  then  letting  P  —  the  total 
wind  pressure,  we  have 

P^  =  ^  (8) 

Inertia. — The  work  done  in  overcoming  the  inertia  of  the  bridge  is  equivalent  to  the 
kinetic  energy  acquired  by  the  total  rotating  mass  when  the  maximum  angular  velocity  has 
been  attained,  and  this  same  amount  of  work  will  have  to  be  done  in  bringing  the  bridge  to 
rest;  although,  in  the  latter  case,  the  resistances  due  to  friction,  and  perhaps  wind,  retard  the 
motion  of  the  bridge  and  assist  in  bringing  it  to  rest. 

In  order  to  determine  the  kinetic  energy  acquired,  it  is  necessary  to  find  the  moment  of 
inertia  of  the  entire  bridge  with  reference  to  its  axis  of  rotation.  However,  it  is  sufficiently 
accurate  to  consider  the  bridge  to  be  a  rectangular  parallelopipedon  of  uniform  density  and  of 
the  same  weight,  whose  length  and  width  are  the  same  as  those  of  the  bridge,  and  whose 
depth,  or  dimension  parallel  to  the  axis  of  rotation,  is  any  convenient  quantit)'. 

To  find  the  mass  which,  placed  at  the  pitch  circle  of  the  rack,  would  produce  the  same 
moment  of  inertia: 

Let  R  —  radius  of  rack  circle ; 
M  —  mass  at  rack  circle; 
/=  moment  of  inertia  of  entire  bridge. 
Then 

Let  V  =  maximum  linear  velocity  at  rack  circle; 
/  =^  time  for  uniform  acceleration  ; 

V 

a  —  rate  of  acceleration  —  —. 

Then  the  kinetic  energy  attained  for  the  velocity  7>  at  the  end  of  /  seconds  =  iA/2'\  and 
the  constant  accelerating  force  applied  at  the  rack  circle  is 

I-       ,^  -^^ 

F^  =  Ma  =  j^^-  (9) 

380.  Summary  of  Resistances  in  Turning.— Summing  the  resistances  at  the  rack 
circle,  we  have 

F=F^  +  F,-]-  F,  +  F^-\-F^  (10) 

which  is  the  force  to  be  applied  at  the  rack  circle  to  overcome  all  the  external  resistances. 

Resistances  of  the  Macliincry. — The  losses  due  to  the  resistances  of  the  moving  parts  of 
the  machinery  are  usually  expressed  in  percentages  of  the  net  resistance  at  the  rack  circle  or  at 
the  ends  of  the  bridge.  These  losses  can  only  be  determined  by  experiment,  but  are  nearly 
a  constant  for  any  given  arrangement  of  machinery.  For  swing  bridges  the  loss  due  to 
resistance  of  the  machinery  is  usually  taken  at  100  per  cent  of  the  work  done  at  the  pitch 
line  of  the  rack  circle  in  turning  the  bridge,  or  at  the  ends  of  the  bridge  in  lifting  the  ends  ; 
then 


IX  Wh\ 

Total  power  required  in  lifting   =  ft.-lbs.  per  second 

'*        "  "       "  turning  =  2Fv  ft.-lbs.  per  second. 


00 


374 


MODERN  FRAMED  STRUCTURES. 


The  time  t  for  lifting,  and  the  time  t  for  acceleration  in  turning,  having  separate  values. 

381.  Design  for  Engine. — For  Lifting  the  Ends. — The  total  horse-power  required, 
including  all  resistances  of  the  machinery,  is 

H-^^=  0^) 

in  which  /  =  time  in  seconds  for  lifting; 
h  —  total  lift  in  feet ; 
W  —  total  load  in  pounds  to  be  lifted. 
For  Turning  the  Bridge. — If  it  is  desired  to  uniformly  accelerate  the  motion  of  the  bridge 
during  the  first  half  of  the  quarter  turn,  and  uniformly  retard  the  motion  during  the  last  half, 
then  /,  the  time  for  acceleration,  becomes  one  half  of  the  whole  time  consumed  in  opening  the 
bridge. 

If  it  is  desired  to  open  the  bridge  by  accelerating  it  until  it  attains  a  given  velocity,  which 
is  to  be  maintained  until  the  motor  is  reversed  or  the  brakes  put  on,  when  the  bridge  is 
brought  to  rest,  then. 

Let  T  =  time  required  for  opening  the  bridge; 
L  =  \  circumference  of  rack  circle  ; 
V  =z  maximum  linear  velocity  at  rack  circle  ; 
/  =  space  on  rack  circle  in  which  bridge  is  to  be  accelerated ; 
/,  =  space  on  rack  circle  in  which  bridge  is  to  be  retarded  ; 
L  —  (/-|-  /J  =  space  on  rack  circle  in  which  bridge  is  to  have  a  uniform  velocity  of  v ; 
t  —  time  for  acceleration  ; 

=  time  for  uniform  velocity  v; 
/,  =  time  for  retardation. 


Then 


*  —      •  t,  —  <  *.  — 


V 


V  =     \  (13) 

When  1=1^  =  —,  which  is  usual,  we  have 
3 

 <'^> 

The  value  of  v  is  then  substituted  in  the  equation  for  the  power  required  in  turning 
(eq.  1 1).    Then  the  total  horse-power  required  in  turning  is 

.    ^■'■  =  fo ('« 

where  F  is  the  total  force  applied  at  the  rack  circle  by  eq.  (10),  and  v  is  the  maximum  linear 
velocity  at  the  rack  circle. 

Having  determined  upon  the  horse-power  to  be  used,  which  determines  the  elements  of 
the  motor,  we  can  then  work  back  from  the  motor  through  the  various  parts  of  the  machinery 
to  the  rack  circle,  or  to  the  ends  of  the  bridge,  and  make  every  part  capable  of  resisting  the 
greatest  force  which  can  come  upon  it  from  the  motor.     In  that  case  it  is  customary  to 


THE  DESIGN  OF  SWING  BRIDGES. 


375 


assume  all  the  parts  to  move  without  any  resistance,  which  is  an  error  on  the  side  of  safety. 
Then  the  total  work  done  by  the  motor  in  foot-pounds  per  second  =  H.P.  X  550. 
Assume  the  motor  to  be  a  steam-engine. 

Let  n  =  greatest  number  of  revolutions  of  engine,  or  maximum  engine  speed,  per  minute  ; 

—  =  length  of  engine  crank  ; 
2 

d  —  diameter  of  cylinder  in  feet ; 

■m  ■=  ratio  of  diameter  of  cylinder  to  stroke,  or  m  —  —  \ 

p  =  steam  pressure  in  pounds  per  square  foot. 
Then  for  an  engine  taking  steam  during  the  full  stroke, 

H.P.  X  550  =  2//-  ^  =  W;  (16) 

^  4  X  60  ^ 


But  d  =  m  X     from  which  knowing      and  assuming  a  value  for  /,  we  can  find  d. 

All  motors  used  for  swing  bridges  should  be  reversible  and  capable  of  exerting  their 
maximum  force  during  any  part  of  a  revolution.  If  the  motor  be  a  steam-engine,  the  engine 
should  be  of  the  double-cylinder  type  *  with  a  link  motion  for  reversal.  The  size  of  the  cylin- 
ders should  be  such  that  the  engine  would  be  capable  of  exerting  the  full  force  required 
during  any  part  of  the  stroke.  With  such  a  motor,  a  swing  bridge  can  be  turned  in  either 
direction,  and  by  reversing  the  motor  it  can  be  used  as  a  brake.  This  is  very  important  in 
bridges  which  have  to  be  opened  or  closed  rapidly,  or  handled  during  high  winds. 

382,  Constants. — The  coefficient  of  rolling  friction,  0, ,  as  determined  in  the  experiments 
on  the  Thames  River  Bridge, f  was  0.003.  The  coefficient  0, ,  for  sliding  friction,  may  be 
taken  at  O.l.  The  coefficient  0,,  for  collar  friction,  maybe  taken  the  same  as  for  sliding 
friction.  In  the  formula  for  the  effect  of  unbalanced  wind  pressure,  /  may  be  taken  as  the 
pressure  due  to  a  wind  velocity  of  thirty  miles  per  hour,  or p  =  X  .004  =  3.6.  However, 
when  the  unbalanced  wind  pressure  is  due  to  a  wind  velocity  of  thirty  miles  per  hour  it  is  the 
predominating  resistance.  Such  an  unbalanced  wind  pressure  may  never  occur  when  the 
bridge  is  swinging  ;  and  it  is  customary,  when  it  is  considered,  to  neglect  all  other  resistances. 
In  other  words,'  the  motor  would  then  be  able  to  hold  the  bridge  only  against  the  unbalanced 
wind,  as  a  brake,  without  attempting  to  turn  the  bridge  against  the  wind. 

Let  P—  total  unbalanced  wind  pressure; 

g—  distance  of  centre  of  pressure  from  the  centre  of  bridge  ; 
R  =  radius  of  rack  circle  ; 

—  force  at  rack  circle  to  balance  P. 

Then    F^  =  ^ 


—  ;  and  the  total  horse-power 


in  which  v  is  the  velocity  at  the  rack  circle  when  the  unbalanced  wind  pressure  occurs.  The 
value  of  V  is  taken  about  one  half  the  maximum  value,  or  when  no  unbalanced  wind  pressure 
is  considered. 

*  This  form  of  engine  is  commonly  jjnown  as  the  marine  engine. 

+  See  a  paper  on  the  Experimental  Determination  of  Rolling  Friction  in  operating  the  Thames  River  Bridge,  in 
Vol.  XXV  of  the  Transactions  of  the  American  Society  of  Civil  Engineers,  by  Alfred  P,  Boiler,  Jr.,  and  H.  J.  Schu- 
macher. 


376 


MODERN  FRAMED  STRUCTURES. 


Design  of  a  21  6-foot  Swing  Bridge. 

383.  Data. — The  following  data  will  be  taken  : 

A  single-track  swing  bridge  216  feet  long,  to  be  operated  by  hand  power  exclusively. 
Two  trusses  placed  16  feet  apart  (14  feet  in  the  clear). 

Four  panels  in  each  arm   25  ft.  o  in.  =  100  ft.  x  2  =  200  feet. 

One  centre  panel   —    16  " 


Total  length  centre  to  centre  of  end  pins  , 

Depth  of  trusses  25  feet  at  ends  (21  feet  clear). 

"       "       "     40  feet  "  centre  ("    "       "  ). 
Distance  from  base  of  rail  to  under  side  of  bridge  not  limited. 
Distance  from  base  of  rail  to  top  of  centre  pier  not  limited. 


=  216  " 


Fig.  402. 

The  links  at  EE',  Fig.  402,  being  at  all  times  practically  vertical,  the  stresses  in  the  two  members  E'e 
are  at  all  times  equal  to  each  other. 

384.  Floor  System  and  Laterals — Having  fixed  upon  the  outline  of  the  trusses,  the  design  of  the  floor 
system,  that  is,  tlie  stringers  and  floor-beams,  follows  as  described  in  Chapter  XXI. 

The  next  step  will  be  to  design  the  wind  trusses,  or  lateral  systems. 


aocdeeacba 

Fig.  403. 

Botiom  Laterals. — Make  all  the  diagonals  of  angles  with  riveted  connections,  so  as  to  resist  both  tension 
and  compression. 

When  it  is  possible  to  do  so,  the  laterals  should  be  riveted  to  the  flanges  of  the  stringers  at  their  inter- 
section. 

The  laterals  in  the  middle  panel  ee.  Fig.  403,  may  or  may  not  be  omitted,  according  to  the  arrangement 
of  the  turntable. 

Top  Laterals  (Fig.  404).  —  Make  all  the  diagonals  of  angles.  Make  all  the  bracing  for  portals  at  B,  D' 
and  E  of  angles.    Make  all  the  struts  at  C  and  E  of  angles. 


D 


i 


Fig.  404. 

When  the  bridge  is  swinging,  the  entire  wind  pressure  on  the  top  lateral  system  goes  to  the  turntable. 
The  wind  forces  accumulating  at  D  are  transmitted  to  the  turntable  by  means  of  direct  bending  in  the  posts 
De  (see  Fig.  402). 

The  laterals  in  the  panel  DE  ave  omitted.  The  lateral  wind  force  at  E  is  transmitted  to  the  turntable 
by  means  of  direct  bending  in  the  posts  E'e. 

When  the  bridge  is  closed  and  the  ends  are  raised,  the  portion  of  the  wind  forces  which  goes  to  the 
abutment  is  transmitted  by  means  of  direct  bending  in  the  end  posts  aB. 

The  stresses  in  the  lateral  systems  are  analyzed  as  shown  in  Chap.  XII.  The  bending  moments  in 
the  posts  are  computed  on  the  assumption  that  the  ends  are  fixed  as  shown  in  Chap.  X,  Art.  151. 
These  bending  moments  are  used  in  proportioning  the  posts,  and  combined  with  the  direct  stresses  from 
the  vertical  loads. 


THE  DESIGN  OF  SWING  BRIDGES 


377 


385.  Dead  Load. — Having  dimensioned  all  of  the  parts  which  are  not  influenced  by  the  stresses  in  the 
main  trusses,  their  weights  may  be  computed.  We  need  therefore  only  assume  the  weight  of  the  main 
trusses.  However,  we  know  that  the  weight  of  the  main  trusses  for  any  swing  bridge  is  about  the  same  as 
for  a  fixed  span  of  the  same  length  minus  the  weight  of  the  turntable  to  be  used. 


We  can  then  assume  for  computations  : 

Total  weight  of  two  main  trusses  =  135,000  lbs. 

"  "  "  floor  system  (computed)  =  70,000  '' 
"        "       "  lateral  systems  (    "       )  =   17,500  " 


Total  weight  of  iron-work  above  turntable  =  222,500 
Check  :  5/''  —  50/  =  total  weight  of  iron-work. 

Assume  dead  load  per  foot  =  w  (iron)  +  400  lbs.  (track). 

The  stresses  are  then  computed  for  the  various  cases  as  shown  in  Chap.  XH,  and  the  members  are  pro- 
portioned as  sliown  in  Chap.  XVII. 

Having  proportioned  the  main  trusses,  their  weight  is  computed,  so  that  we  finally  get  the  computed 
weights  of  all  the  ironwork  above  the  turntable. 

386.  The  Turntable,  including  the  turning  and  end-lifting  arrangements. 

Let  us  assume  that  the  total  weight  of  the  iron-work  above  the  turntable  came  out  as  assumed,  5/"  —  50/ 
—  222,500  lbs. 

Weight  of  iron-work  above  turntable  =  222,500  lbs. 
"       "  track  =  216  X  400  —  86,400  " 

Total  dead  load  on  turntable    =  308,900  " 
Maximum  live  load  on  turntable  (Case  IV)  =  (say)  405,000  " 

Total  live  load  -|-  dead  load  on  turntable    =  "     713,900  " 

The  turntable  to  be  used  is  to  be  rim-bearing  and  centre-bearing  combined.  (See  Fig.  405  for  general 
outline  of  turntable.)  The  total  load  on  the  turntable  comes  on  the  four  corners  P  of  the  square  formed  by 
the  longitudinal  and  transverse  girders,  as  explained  in  Art.  375  of  this  chapter. 

r 

If  P  is  the  load  on  each  corner,  then  one  half  goes  each  way  to p,  so  that  the  load  at  p  is  — .    The  load 

2 

at  p  is  carried  equally  to  the  two  adjacent  radial  girders  by  means  of  short  cross-girders  aa,  called  dia- 
phragms. The  load  at  a  is  then  again  divided,  a  portion  going  to  the  centre  C  and  the  remainder  to  the 
sixteen  points    on  the  rim.    For  this  case 

the  load  on  the  rim 
and  the  load  on  the  centre 

The  longitudinal  and  transverse  cross-girders  are  designed  as  simple  beams  supported  at  and  loaded 
at  each  end  by  a  load  of  \P.  The  diaphragms  aa  are  designed  as  beams  supported  at  each  end,  and  loaded 
at  the  middle  by  a  load  of  \P. 

P 

The  radial  girders  Cd  are  designed  as  beams  supported  at  each  end,  and  loaded  by  a  load  of  —  at  2.5 

4 

feet  from  one  end. 

In  designing  the  circular  girder  or  drum,  the  analysis  for  a  straight  beam  will  not  apply.  However,  it  is 
assumed,  in  order  to  provide  ample  strength  and  stiffness,  that  the  circle  is  composed  of  a  series  of  straight 
beams  of  a  length  equal  to  two  segments  dd,  or  one  eighth  of  the  circumference  of  the  circle.  These  beams 
are  assumed,  furthermore,  to  be  disconnected,  that  is,  free  at  their  ends,  and  loaded  in  the  middle  by  a  load 


P  8.8 

=  —  X  ,  for  this  case,  with  end  supports. 

4  11-3 

Total  dead  load  on  turntable  =  308,900  lbs.; 

Maximum  live  load  on  turntable  -    405,000  " 

Turntable  weighs  (say)  =    86,300  " 


\P  X  8.8 

4P  X  2.5 
II-3 


=  31  ISA 
-  o.SSsA 


MODERN  FRAMED  STRUCTURES. 


Fl&  40$. 


THE  DESIGN  OF  SWING  BRIDGES, 


379 


395,200 

Total  dead  load  on  rollers  =  x  3.1 15  =  say  307,800  lbs.; 

4 

405,000 

«   live  load     "     "   =  — -x  3.115=  "  315,400  " 

4 

•?Q5,200 

"   dead  load  "  centre  =  ~        x  0.885  =  "    87,400  " 

4 

4.05,000 

"   live  load    "     "   =   x  0.885  =  "     89,600  " 

4 

Assumed  diameter  of  rollers  =18  inches ; 

Number  of  rollers  =  ^^"^^  ^  ^  =  say  40  rollers  18  in.  diameter> 

1-5 

Permissible  bearing  on  rollers  (static)  =  300  x  18  =  5400  lbs.  per  linear  inch; 

"  "       "       "     (moving)..  =  150  x  18  =  2700    "     "       "  " 

62  x  200 

Total  number  linear  inches  required  (static)  =      —  =  124  inches; 

5000 

307,800  .  , 

"        "         "         "          "     (movmg)  =   =  123  mches. 

^         ^        2500  ^ 

Use  forty  rollers,  18  in.  diameter.  4  in.  face  =  160  linear  inches,  to  be  of  cast-iron  with  chilled  faces. 
The  treads  for  rollers  should  be  made  of  wrought-iron  or  steel.  The  latter  is  preferable  and  may  be  of  the 
same  grade  as  that  used  for  track  rails  for  railways. 

Permissible  bearing  on  centre  (static)  =  6000  lbs.  per  square  inch ; 

"  "       "      "      (turning)  =  3000  ' 

A                                     .                                 177,000  .  , 

Area  required  m  disks  (static)  =  —  —  29.5  square  inches; 

6000 

87,400 

 '     (turning)  =  _irt_  =  29.2      "  " 

3000 

Use  a  centre  pin  6 J  inches  diameter  =  33.2     "  " 

"    3  disks        6\     "  "   =  30.7      "  " 

The  upper  and  lower  disks  to  be  hard  steel  and  the  middle  disk  to  be  phosphor-bronze.    The  centre 

177,000 

casting  requires  for  bearing  on  the  masonry  =  590  square  inches.    Use  casting  with  base  27.5  inches 

300 

diameter  =  594  square  inches. 

The  next  step  in  order  should  be  the  turning  and  end-lift  arrangements;  but  before  we  can  proceed 
with  the  latter  it  is  necessary  to  know  the  end  deflection. 

387.  Framing  of  the  Trusses  for  Proper  Camber  and  End  Deflection — The  lengths  of  the  members 
should  be  such  that  with  a  proper  elevation  of  the  ends  the  bottom  chord  is  a  horizontal  line  under  the 
maximum  load. 

(I.)  Compute  the  normal  lengths  of  the  members,  assuming  the  elevation  of  the  ends  to  be  normal. 
(II.)  Increase  or  diniinisli  the  lengths  of  the  members  according  as  they  are  under  compression  or 

tension,  respectively,  when  the  full  live  load  is  on  the  bridge.    This  change  of  length  is  rigidly  —  for  each 

E 

member,  where  /  is  the  total  live  and  dead  load  stress  per  square  inch  in  that  member.  Since  such  meas- 
urements of  length  are  made  only  to  the  nearest  ^  inch,  the  change  of  length  used  is  taken  so  as  to  give 
the  final  length  in  even  thirty-seconds  of  an  inch. 

(III.)  Find  the  change  produced  in  the  elevation  of  the  ends  due  to  the  change  in  the  lengths  of  the 

members  by  the  formula  given  in  Art,  200,  Chap.  XV,  D'  =  wherein"^ is  to  be  taken  as  the  "change 

of  length  used  "  in  (II)  for  each  member.    The  value  of  D'  in  this  case  is  called  the  camber  deflection. 

(IV.)  Find  the  end  deflection,  span  swinging,  dead  load  only  acting,  by  the  formula     =  2^^.  The 

E 

value  of  D  in  this  case  is  called  the  true  deflection. 


38o 


MODERN  FRAMED  STRUCTURES. 


(V.)  Assume  that  one  half  of  the  deflection,  from  (IV),  is  eliminated  by  lifting  the  ends  ;  then  shorten 
the  upper  chord,  near  the  centre  of  the  bridge,  so  as  to  raise  the  arms  the  remaining  one  half  of  the  end 
deflection  from  (IV)  plus  the  whole  deflection  caused  by  the  change  in  the  length  of  the  members,  or  con- 
dition (III).  Let  Z>  =  deflection  from  (IV),  or  the  true  deflection;  let  Z*' =  deflection  from  (III),  or  the 
camber  deflection. 

Assume  that  \D  is  eliminated  by  lifting  the  ends  ;  then  the  remaining  \D  +  D'  is  to  be  eliminated  by 
shortening  or  lengthening  the  upper  chord  near  the  centre  of  the  bridge,  according  as  the  value  of  \D  +  D' 
is  plus  or  minus.  However,  although  the  value  of  D'  is  usually  minus  or  an  upward  deflection,  it  is  nevei 
greater  than  \D  \  so  that  \D  +  {  —  D')  is  always  plus,  which  necessitates  shortening  the  upper  chord,  and 
never  lengthening  it. 

388.  The  Turning  Arrangements — Let  us  assume  the  time  for  opening  or  closing  the  bridge  by  hand 
power  to  be  240  seconds;  then  if  the  time  for  uniform  acceleration,  /,  =  120  seconds,  the  maximum  velocity 

.  25  X  3.14 

at  the  rack  circle,  which  is  twice  the  mean  velocity,  v,  =  =  0.16  foot  per  second.    If  the  bridge  is 

4  X  120 

to  be  operated  exclusively  by  hand  power,  no  provision  is  made  for  swinging  the  bridge  against  unbalanced 
wind  pressures ;  so  that  the  force  F^  at  the  rack  circle  necessary  to  overcome  wind  is  here  neglected. 
From  eq.  (10)  we  have 

F  —  Fr  +  Fs  +  Fc  +  Fa  =  force  at  rack  circle ; 

Fr=  (pi  =  0.003     307,800  X   =  831  lbs.; 

d  o.z 

Fc  =  (p3  IV--  =  o.  I  X  87,400  X  —  =   116  " 

ZR  3  X  12.5 

„     4    rs'  — n'    4>i^r     4     26  0.1x307,800 

-f;  =  -  X  —          x  — -  =  ?  X  -  X   =  860  " 

3     r^  —  r-i         R        38        12.5  X  12 

395. 200  X  62.5'  X  0.16 

R^t       32.2  X  12.5  X  120   

F  =  2223  " 

The  total  force  required  to  overcome  all  resistances  equals  twice  the  net  resistance  =  4446  lbs.  With 

4446 

two  men  on  the  turning  lever  at  50  lbs.  each,  the  multiple  of  force  by  gearing  =  =  44.46.    The  time 

2  X  50 

to  open  or  close,  with  men  walking  at  the  uniform  rate  of  4  ft.  per  second  =  44.46  x         x  '  =  222  seconds 

4  4 

=  3  minutes  42  seconds. 

The  maximum  rate  of  walking  would  then  be  8  feet  per  second.  If  the  velocity  v  is  generated  in  less 
than  120  seconds,  the  maximum  rate  of  walking  would  be  less,  to  open  in  the  same  time. 

389.  The  End-lifting  Arrangements. — The  total  power  required  in  lifting  the  ends,  eq  (11),  =  2^^-^^^ 
in  foot-pounds  per  second,  in  which 

W  =  total  dead  load  above  turntable  =  308,900  lbs. ; 

h  =  lift  in  feet  =0.1  ft. ; 

/  =  time  required  for  lift  in  seconds. 

2  X  3  X  308,900 

With  two  men  on  turning  lever  at  50  lbs.  each,  the  multiple  of  force  by  geanng  =  —  ■  =  1158. 

*  16x2x50 

The  time  required  to  raise  the  ends,  with  men  walking  4  ft.  per  second,  =  1158  x  o.  i  x  i  =  29  seconds. 
In  case  it  should  become  necessary  to  operate  the  bridge  more  rapidly,  it  would  be  advisable  to  intro- 
duce a  steam-engine  or  some  other  form  of  motor. 

Assume  the  time  is  to  be  limited,  so  that  the  total  time  for  opening  or  closing  shall  not  exceed  75  sec- 


THE  DESIGN  OE  SWING  BRIDGES. 


381 


onds,  including  the  time  necessary  for  raising  or  lowering  the  ends.  Let  the  time  of  opening  be  60  seconds 
and  the  time  for  acceleration  =  30  seconds ;  then  the  maximum  velocity  at  rack  circle, 

25  X  3.14  .    .  . 

V,  =  —   =  0.6;  ft.  per  second ; 

4  X  30 

=  395,200  X  6^5- X  0.65  ^  ggg^  . 

y?v       32.2  X  12.5  X  30 

Fr-\-  Fs-V  Fc  as  before   =  1807  " 

F  = 


The  force  required  at  the  rack  circle  for  such  a  rapid  acceleration  will  be  found  to  be  ample  to  hold  the 
bridge  against  any  unbalanced  wind  pressure,  so  that  the  wind  is  here  again  neglected. 

.    ,  2/^2/     2  X  8468  X  0.65  „ 

Horse-power  required  =  ■  =   =  20  H.  P.; 

550  550 


Let  t,  the  time  for  lifting,  =  5  seconds. 


^  i6i        2  X  ■}  X  308,900  X  o.i       _  TT  D  . 

Horse-power  required  =    =   =  4.2  H.  P.; 

550  16  X  550  X  5 

Entire  time  required  for  opening  or  closing  —  65  seconds. 

Use  20  horse-power  engine  with  double  cylinder  and  link-motion  for  reversal. 

Then,  from  eq.  (16),  H.  P.  x  550=  11,000  foot-pounds  per  second. 

From  eq.  (17)  we  have 


m  =  y  — 


000  X  I20_ 


in  which  /  =  pressure  in  pounds  per  square  foot ...   =  say  5760  lbs. ; 

/  =  length  of  stroke   -  say  0.7  ft.; 

»  =  maximum  engine  speed  or  revolutions  per  minute  =  250. 


.  /         11,000  X  120 

»i  =  y   _  _  =  0.91. 


Then 

5760  X  3. 14  X  0.343  X  250 
Diameter  of  cylinders  =  m  x  I  =  say  7  inches. 

250  X  8  X  2 

Two  cylinders  7  in.  diameter,  8  in.  stroke,  with  a  maximum  piston  speed  of   =  say  333  feet 

per  minute,  which  is  below  the  maximum  piston  speed  generally  allowed. 

After  determining  the  elements  of  the  motor  to  be  used,  the  next  step  would  be  to  design  the 
machinery,  making  every  part  strong  enough  to  resist  the  maximum  force  which  can  come  upon  it  from  the 
motor.  This  branch  of  the  design  of  swing  bridges  will  not  be  treated  here,  as  it  properly  comes  under  the 
head  of  machine  design,  and  is  fully  treated  in  standard  books  on  this  subject. 


38a 


MODERN  FRAMED  STRUCTURES. 


CHAPTER  XXV. 
TIMBER  AND  IRON  TRESTLES  AND  ELEVATED  RAILROADS. 

I.  Timber  Trestles. 

390.  Use  of  Timber  Trestles. — Timber  trestles  are  used  in  America  under  the  follow- 
ing conditions : 

{a)  When  the  first  cost  is  less  than  that  of  an  earth-fill. 

ip)  When  water-way  must  be  left  and  it  is  not  practicable  to  build  either  bridges  or  cul- 
verts on  the  first  construction  of  the  road. 

{c)  When  the  financial  condition  of  the  corporation  will  not  admit  of  a  more  permanent 
kind  of  structure. 

It  has  been  estimated  that  there  are  at  present  in  the  United  States  at  least  2400  miles 
of  railway  wooden  trestle,  of  which  at  least  800  miles  are  likely  to  be  permanently  maintained 
as  such.  It  is  very  common  to  build  wooden  trestles  on  a  new  line  of  road,  and  when  it 
needs  renewal  to  make  an  earth-fill.  This  can  be  done  very  cheaply  after  the  line  is  con- 
structed ;  and  if  the  earthwork  comes  high  on  the  first  construction  of  the  line,  it  may  be 
economical  to  do  this.  Wooden  trestles  are  built  of  all  heights  up  to  150  feet,  and  although 
a  multitude  of  patterns  have  been  successfully  employed,  there  are  certain  guiding  principles 
which  lead  to  a  nearly  uniform  best  practice.  No  discussion  will  be  here  given  of  simple  pile 
bents,  as  not  properly  being  framed  structures.  Neither  will  the  subject  of  pile  driving  or  the 
bearing  resistance  of  piles  be  here  entered  upon.*  The  foundation  of  a  framed  timber  bent 
is  preferably  composed  of  masonry  or  solid  rock,  but  next  to  this  piles  are  to  be  preferred. 

391.  The  Framed  Bent. — Some  illustrations  of  properly  framed  timber  bents  are  shown 
in  Plate  VI,  those  in  the  upper  portion  showing  the  practice  of  the  Pennsylvania  Railroad 
Company,  and  in  the  lower  portion  that  of  the  Norfolk  &  Western  Railway.  There  are 
always  two  vertical  struts  and  two  or  more  inclined  ones,  called  "batter  posts."  The  batter 
is  2\  in.  to  3  in.  to  one  foot.  These,  together  with  the  sill  and  cap  and  the  diagonal  braces 
make  up  the  frame,  or  '*  bent "  as  it  is  called,  which  is  the  elementary  form  of  all  trestle- 
work.  When  the  timbers  are  all  in  pairs  it  is  called  a  "  double  bent"  see  (the  pattern  used  by 
the  Toledo,  St.  Louis  &  Kansas  City  Railroad,  Plate  IX).  As  in  the  case  of  a  Howe  truss 
bridge,  the  timbers  here  are  not  nicely  proportioned  for  their  loads,  but  the  sizes  are  made 
large  to  allow  for  decay.  Purely  empirical  rules  of  practice  are  followed  in  this  matter,  and 
the  various  types  shown  in  Plates  V  to  X  have  been  selected  as  offering  good  examples  to 
follow. 

392.  Bearing  Joints. — In  Plate  VI  two  kinds  of  bearing  joints  are  shown.  The  mortise 
and  tenon  (Pennsylvania  Railroad  Standard)  is  giving  place  to  the  split  cap,  shown  in  the 
Norfolk  &  Western  plans,  Plate  VI.  Here  the  posts  are  notched  to  receive  the  two  cap  or 
sill  pieces,  a  smooth  surface  being  made  on  the  outside  at  top  to  receive  the  stringer,  and 
at  bottom  to  rest  on  the  bearing  timbers.  The  advantages  of  this  latter  joint  are  that  it 
facilitates  repairs,  and  secures  better  timber  for  the  caps  and  sills,  since  they  are  only  half  the 
thickness. 


*  See  Baker's  "  Masonry,"  John  Wiley  &  Sons,  for  the  best  treatise  on  Foundations,  including  piles. 


TIMBER  TRESTLES. 


383 


Another  form  of  this  joint  is  that  shown  in  Plate  IX,  where  all  the  bearings  are  made  on 
cast-iron  plates.    This  adds  materially  to  the  life  of  the  joint 
and  insures  a  more  uniform  distribution  of  the  pressure. 

Another  method  of  accomplishing  the  same  thing  is  by 
means  of  wrought-iron  plates,  as  shown  in  Fig.  406.  This 
joint  has  been  used  by  the  New  York,  Lake  Erie  &  Western 
Railway,  and  is  more  fully  described  in  Engineering  Neivs  for 
Nov.  5,  1887.  The  plates  may  be  stamped  out  and  spiked 
on.  They  would  probably  more  than  save  their  cost  in  the 
time  saved  on  the  ground  in  the  framing  which  is  here  ren- 
dered unnecessary. 

In  all  cases  where  timber  is  used  in  compression  across 
the  fibres,  attention  must  be  given  to  the  area  presented  to 
the  imposed  load,  using  the  working  stresses  given  on  p.  354. 

393.  Floor  Sy'sX.^rciS.Stringers. — The  stringers  should 
consist  of  two  or  three  sticks  each,  well  separated  from  each 
other  by  cast-iron  washers,  and  bolted  together.  They  should 
extend  over  two  spans  and  terminate  at  alternate  bents.  They 
should  never  be  notched  down  upon  the  caps.  This  greatly 
weakens  the  stick  as  a  beam,  causing  it  to  fail  by  shearing, 
or  splitting  back  from  the  base  of  the  notch.  They  should  be 
spliced  by  means  of  large  packing  blocks,  which  are  notc4ied 
down  over  the  cap  and  bolted  to  the  stringers,  as  shown  in 
Plate  VI.  On  the  Minneapolis  &  St.  Louis  Railway  (Plate  V) 
these  packing  pieces  are  simply  cuUings  from  the  stringer 
pieces,  sawed  to  5^-foot  lengths,  and  inserted  with  plain  cast-iron  washers  for  air  circulation. 
The  stringers  are  held  laterally  upon  the  caps  either  by  bolting  through  or,  better,  by  cast- 
iron  brackets,  as  shown  in  Plate  V. 

Corbels  should  not  be  used.  They  add  to  the  cost,  and  furnish  large  joints  for  the  storage 
of  water,  and  hasten  decay.  They  are  shown  on  Plates  V  and  VIII,  but  they  should  be  omitted 
from  all  wooden-trestle  construction.  The  bearing  area  and  beam  strength  should  be  suffi- 
cient without  them.  Thus  in  case  of  a  span  of  16  feet,  with  an  equivalent  live  and  dead  load 
of  6000  lbs.  per  foot,  or  of  48,000  lbs.  coming  to  the  cap  from  one  side,  there  should  be  at 
least  three  8-inch  stringers,  and  if  the  cap  is  12  inches  wide  the  bearing  area  is  6  X  24  =  144 
square  inches.  This  gives  a  pressure  of  330  lbs.  per  square  inch,  which  is  about  the  limit  of 
allowable  bearing  s'tress  on  white  pine  across  the  grain,*  and  might  be  permitted.  If  yellow 
pine  were  used  for  this  work,  three  6-inch  sticks  would  be  sufficient. 

For  very  high,  and  therefore  expensive,  bents  they  should  be  placed  farther  apart,  as  in 
Plate  VII,  and  the  stringers  braced  out  from  the  bent  as  shown.  The  shortening  of  the  span 
by  means  of  corbels,  as  in  Plate  V,  is  of  doubtful  economy  or  expediency.  Neither  is  it  ad- 
visable to  truss  the  stringers  with  iron  rods  and  king-posts,  as  is  sometimes  done.  The  addi- 
tional strength  from  this  source,  with  shallow  depths  to  the  trussing,  is  very  small,  and  too 
great  reliance  is  apt  to  be  put  in  such  a  combination. 

Packing. — The  stringers  should  never  be  more  than  8  inches  thick,  and  are  usually  from 
14  to  18  inches  deep.  They  should  be  separated  two  inches  or  more  apart,  and  packed  with 
the  splicing  timbers  at  the  bents,  as  shown  in  Plates  V  and  VI.  These  splice  or  packing  pieces 
are  notched  over  the  cap  timber,  but  the  stringers  are  not  notched  or  framed  in  any  way. 
When  corbels  are  used  the  packing  timbers  may  be  dispensed  with,  and  cast-iron  spools  or 
separators  used  for  packing,  as  shown  in  Plate  V. 

*  See  p.  354,  for  working  stresses.  If  smaller  caps  are  used,  a  sufficient  bearing  area  demands  the  use  of  corbels 
which  should  be  made  of  white  oak  or  some  other  durable  hard  wood.    See  Plate  Xa. 


384 


MODERN  FRAMED  STRUCTURES. 


The  ties,  guard-rails,  safety  or  rerailing  devices,  etc.,  are  not  peculiar  to  this  form  of 
structure. 

394.  Sway  Bracing. — These  are  designed  to  stay  the  bents  against  lateral  forces  of  all 
kinds  and  to  add  to  the  lateral  rigidity  of  the  structure.  The  batter  posts  assist  greatly  to 
this  end,  but  when  the  bents  are  more  than  18  or  20  feet  high  some  kind  of  sway  bracing 
should  be  employed.  This  bracing  usually  consists  only  of  2-inch  to  4-inch  plank,  from  8 
to  12  inches  wide,  spiked  to  the  sides  of  the  bents,  as  shown  in  the  plates.  This  is  a  very 
unscientific  and  inefficient  kind  of  brace.  This  method  of  fastening  cannot  possibly  develop 
the  full  strength  of  the  brace.  If  the  brace  is  thick,  so  as  to  have  strength  as  a  strut,  it  makes 
the  leverage  on  the  spikes  so  great  that  their  strength  is  greatly  reduced.  If  the  brace  is  thin 
it  has  little  strength  as  a  strut,  and  besides  the  spikes  have  little  bearing  area.  It  is  not 
practicable  to  insert  simple  struts  in  the  plane  of  the  bent,  as  "  bridging,"  since  this  would 
require  too  many  joints  and  too  much  cutting  and  fitting.  The  ordinary  method  of  cover- 
planks,  attached  on  the  two  faces  of  the  bent,  seems  to  be  the  only  feasible  scheme.  It  now 
remains  to  provide  the  most  efificient  connection.  The  greatest  objection  to  the  use  of  spikes 
is  the  small  bearing  area  on  the  wood.  Large  treenails  (wooden  pins)  would  be  much  better. 
If  3x6-inch  plank  be  used,  and  treenails  \\  inches  in  diameter,  then  the  bearing  area  is  4^ 
square  inches  and  the  shearing  area  of  the  pin  is  if  square  inches.  If  the  plank  be  of  white 
pine  and  the  treenail  of  long-leaf  yellow  pine  or  of  white  oak,  the  strength  of  the  joint  would 
be  very  much  greater  than  if  it  were  made  of  one  3X  12-inch  plank  fastened  with  three  -J  inch 
square  ship-spikes.  The  spikes  will  also  rust  rapidly  just  at  the  joint  where  the  strength  is 
most  required.  The  treenails  should  be  turned  to  a  uniform  diameter  about  of  an  inch 
greater  than  that  of  the  hole.  The  holes  in  the  braces  could  be  bored  before  erection,  and 
then  the  braces  lightly  spiked  to  place,  when  the  holes  in  the  posts  and  caps  could  be  bored 
in  exact  line  with  the  outer  holes.  This  would  make  a  reasonably  rigid  and  permanent  joint, 
which  would  develop  its  full  resistance  without  appreciable  distortion. 

When  the  bents  are  more  than  about  20  feet  high  they  should  be  divided  into  panels  or 
stories,  and  sway  bracing  introduced  into  each  panel.  It  is  not  possible  to  compute  the  stress 
on  the  sway  bracing,  except  for  assumed  wind  pressures.  In  fact,  it  never  is  computed  except 
for  very  high  structures.  On  low  trestles  the  sway  bracing  is  designed  to  resist  the  lateral 
throw  of  the  engine  and  train  rather  than  the  wind  pressure. 

395.  Longitudinal  Braces. — It  is  not  uncommon  to  merely  join  the  bents  together  by 
outside  horizontal  girts,  or  waling-strips,  as  shown  on  Plate  IX,  the  diagonal  braces  being 
entirely  omitted.  This  is  a  grave  omission.  For  short  reaches  of  low  trestling  it  may  be 
allowable  to  omit  these  diagonal  braces  between  the  bents,  where  there  are  provided  substan- 
tial abutment  resistances  to  take  up  the  pull  or  thrust  of  the  train  ;  but  it  is  not  wise  to  rely 
upon  such  abutment  action. 

The  pull  of  two  locomotives  at  the  head  of  a  train  may  be  as  much  as  forty  or  fifty 
thousand  pounds.  When  these  are  upon  a  stretch  of  trestle-work,  something  must  resist  this 
longitudinal  external  force.  When  braking  a  train  also,  the  horizontal  thrust  may  be  much 
greater  than  this,  and  these  great  longitudinal  pulls  or  thrusts  should  be  amply  provided  for. 

In  this  case  a  series  of  direct  end-bearing  struts  may  be  introduced,  as  is  done  on  the 
Pennsylvania  Railroad  design  in  Plate  VI.  Here  a  single  row  of  8"  X  8"  braces  are  intro- 
duced, joining  alternate  top  cap-sills  with  the  intermediate  bottom  sills  in  single-story  trestle- 
work,  or  with  the  intermediate  cross-girts  in  multiple-story  trestles,  as  in  Plates  VIII  and  IX. 
With  this  joint  there  is  a  series  of  direct  lines  of  struts  from  top  to  bottom,  all  end-bearing, 
as  any  strut  should  be.  There  is  no  hindrance  to  inserting  these  braces  in  this  manner.  It 
is  very  bad  practice,  therefore,  to  insert  them  in  any  other  way,  as  is  done  in  Plates  VII 
and  IX,  where  they  are  simply  spiked  or  bolted  on  the  outer  sides;  and  to  omit  them  alto- 
gether, as  is  so  often  done,  even  in  high  trestle-work,  is  simply  criminal. 


TIMBER  TRESTLES, 


385 


Standard  Pile-trestle,  Chicago  &  Northwestern  Railway. 


386  MODERN  FRAMED  STRUCTURES. 

Plate  VI. 


Scale  for  trestles. 
Standard  Framed  Trestles,  Pennsylvania  Railroad. 


Standard  Trestles,  Norfolk  &  Western  Railroad. 


Elevation  High  or  Multiple  Story  Trestle. 


388  MODERN  FRAMED  STRUCTURES. 

Plate  VIII. 


SCALE  OF  FEET 

Former  Standard  Trestle,  CHAKLEbTON,  Cimcinnati  &  Chicago  Railroad. 


Plate  Xa. 


;3STRS.L0liGLEAFPi«it 
'3Cof)BEL3.WjHITE0«"^ 
jjy""'  Kap.  White  0*r 


Posts. Reo  CrPREss. 


■DRIP  HOLES 


2tX6x4- 


-SillS. 
J  White  Oak 


Scale- f  Inch  -  I  Toot. 
Fig.  I. 


STRsLOHtLEAfPlnt' 

CAPS.VVniTt  0»n. 


Posts, RtoCrpRtss 


_D££P  \  \  .Sills.. 

°^  I  White  Oak 7 


Scale:  i-  Inch  =/  rooTI:, 
Fig.  2. 


_4Strs.LonsLeafPini 
^Caps.White  Oak 


Sills 


ScAL£  f  Inch '1  Foot. 
Fig.  3. 


ScALe!tli*CH-IF<f$Zi . 

Viz  4. 


530(1 


MODERN  FRAMED  STRUCTURES. 


The  ordinary  practice  in  the  design  of  timber  trestles  has  been  investigated  by  Mr.  A.  L. 
Johnson,*  chief  computer  on  the  U.  S.  Timber  Test  work,  and  he  has  found  that  the  current 
factors  of  safety  in  such  structures  vary,  for  different  members  and  for  different  '  inds  of  stress, 
from  less  than  unity  to  25,  the  former  being  for  crushing  or  bearing  stress  across  the  grain. 
In  other  words,  these  structures  are  now  very  irrationally  designed.  He  recommends  the  two 
designs  in  Figs,  i  and  2,  Plate  Y^a.  Here  are  given  both  the  species  and  the  sizes  best  suited 
to  the  several  parts,  the  design  in  Fig.  i  being  with  corbels  and  that  in  Fig.  2  without  them. 
In  these  designs  the  following  factors  of  safety  were  used : 

Stringers  in  cross-breaking  , , ,  "  -\- 

Stringers  in  deflection,       span  ,   2  -f- 

Stringers  in  end-bearings   4  ~h 

Cap  in  bearing  value   3  -|- 

Fosts  as  columns  ...   7.4  -j- 

Mr.  G.  Lindenthall,  in  commenting  on  the  designs,  proposes*  the  designs  shown  in 
Figs.  3  and  4,  PI.  Xr?,  these  being  slight  modifications  of  Mr.  Johnson's  designs.  Here  all 
mortises  are  omitted,  wooden  doM-^el-pins  and  splice-planks  being  used  in  their  place.  Mr. 
Lindenthall  strongly  recommends,  also,  the  covering  over  of  the  stringers  with  galvanized 
sheet-iron  to  protect  them  from  the  weather  and  from  the  lodgment  of  live  cinders,  which 
causes  the  destruction  of  so  many  wooden  trestles  by  fire.  Mr.  Bouscaren  has  tried  planking 
over  the  structure  and  covering  it  with  gravel,  with  good  results. 

The  following  table  of  working  unit  stresses  is  based  on  factors  of  safety  of  10  in  tension, 
5  in  longitudinal  compression,  4  in  lateral  compression,  6  in  cross-breaking,  2  in  modulus  of 
elasticity,  and  4  in  shearing,  and  on  the  most  reliable  information  obtainable  to  date; 


AVERAGE  SAFE  ALLOWABLE  UNIT  STRESSES  IN  POUNDS  PER  SQUARE  INCH  RECOMMENDED 
BY  THE  COMMITTEE  ON  "STRENGTH  OF  BRIDGE  AND  TRESTLE  TIMBERS,"  AMERICAN 
ASSOCIATION  OF  RAILWAY  SUPERINTENDENTS  OF  BRIDGES  AND  BUILDINGS,  FIFTH 
ANNUAL  CONVENTION,  NEW  ORLEANS,  OCTOBER,  1895. 


Kind  of  Timber. 


Factor  of  safety. 


White  oak  

White  pine  

Southern,  long-leaf,  or  Georgia  ycl 

low  pine  

Douglas,  Oregon,  and  Washington 
fir  or  pine  : 

Yellow  fir  

Red  fir  

Northern  or  short-leaf  yellow  pine. 

Red  pine  

Norway  pine  

Canadian  (Ottawa)  white  pine  

Canadian  (Ontario)  red  pine  

Spruce  and  Eastern  fir  

Hemlock  

Cypress  

Cedar  

Chestnut  

California  redwood  

California  spruce  


Tension. 

Compression. 

Transverse  Rupture. 

Shearing. 

With 
Grain. 

Across 
Grain. 

With  Grain. 

Across 
Grain. 

Extreme 
Fibre 
Stress. 

6 

Modulus 
of 

Elasticity. 

With 
Grain. 

Across 
Grain. 

End 
Bearing. 

Columns 
under 

15  Diam- 
eters. 

10 

10 

5 

5 

4 

2 

4 

4 

1,000 
700 

1,200 

1,200 
1,000 
900 
900 
800 
1,000 
1,000 
800 
600 
600 
800 
900 
700 

200 

50 

60 

1,400 
1,100 

1,600 
1,600 

goo 

700 

1,000 
1,200 

500 
200 

300 

1,000 

700 

1,200 

1,100 

800 
1,000 
800 
700 

550,000 
500,000 

850,000 
700,000 

200 
100 

150 
150 

1,000 
500 

1,250 

50 
50 

1,200 
1,200 
1,200 

800 
800 
800 
1,000 
1,000 
800 
800 
800 
800 
1,000 
800 
800 

250 
200 
200 

600,000 
600,000 
600,000 

100 

1,000 

100 

I09 
100 
100 

800 
700 
600 
800 
800 
800 
750 
800 

700,000 
600,000 
450, 000 
450,0.10 
350,000 
500,000 
350,000 
600,000 

50 

1,200 

200 
150 
200 
200 
250 
200 

750 
600 

1,200 
1,200 

400 
400 

150 
100 

*  See  Bulletin  No.  12,  Forestry  Division  U.  S.  Agricultural  Department,  1896. 


IRON  TRESTLES. 


39' 


396.  Bent  and  Post  Splices. — When  the  trestle  is  higher  than  about  forty  or  fifty  feet 
it  becomes  necessary  to  splice  the  posts.  There  a^e  several  ways  of  doing  this.  Entirely 
separate  bents  can  be  constructed  in  sections  or  stories,  and  set  one  on  top  of  another,  as 
shown  in  Plate  X,  with  an  air  space  between  the  sills,  except  under  the  posts,  where  seasoned 
oak  bearing-planks  are  carefully  framed  in. 

Or  the  posts  may  abut  end  to  end,  with  side  notches  cut  at  the  joints  for  cross-girts,  as 
shown  in  Plate  VII. 

Or  cast-iron  bearing  caps  can  be  employed  as  shown  in  Plate  IX,  these  resting  on  one 
common  sill,  or  cross  girt,  at  each  splicing  section.  All  these  are  good  joints  when  well 
executed,  but  the  last  named  is  probably  the  nost  lasting,  and  the  one  which  settles  or 
crushes  down  the  least. 

397.  Working  Stresses. — The  safe  working  stress  per  square  inch  for  different  timbers 
for  different  kinds  of  resistance  is  given  pretty  fully  in  Chapter  XXIII,  on  Howe  Trusses. 
As  there  shown,  it  is  important  to  load  timber  in  compression  longitudinally  only.  So  in  a 
timber  trestle  it  would  be  best  to  avoid  crushing  the  timber  across  the  grain  at  all  if  possible, 
or  at  any  rate  introduce  as  small  an  amount  of  timber  under  this  kind  of  stress  as  possible, 
Thus  in  the  Norfolk  &  Western  Railroad  plans  in  Plate  VII,  the  posts  are  given  end  bearings 
continuously  from  top  to  bottom,  being  simply  notched  at  the  joints.  On  the  other  hand, 
Plate  VIII  shows  24  inches  of  timber  under  a  lateral  crushing  load  each  25  feet  in  height. 

II.   Iron  Trestles. 

398.  General  Design. — Iron  railway  trestles  for  single  track  usually  consist  of  a  series 
of  plate-girder  spans  supported  on  iron  columns,  as  shown  in  Fig.  407.    The  columns  are 


Fig.  407. 

braced  together  transversely  in  pairs  to  form  bents,  as  shown  in  the  cross-section.  Fig.  407. 
The  bents  are  then  braced  together  in  pairs  to  form  towers.  The  towers  are  designed  so  as 
to  have  sufficient  stability,  both  longitudinally  and  transversely,  to  withstand  any  force  which 
may  tend  to  overturn  them.  The  only  force  that  would  tend  to  overturn  them  in  a  trans- 
verse direction  on  a  straight  track  is  the  pressure  of  the  wind  on  the  structure  and  on  the 
train.  The  friction  of  the  wheels  on  the  rails  caused  by  suddenly  applying  the  brakes  to  a 
moving  train  is  the  force  which  tends  to  overturn  the  towers  in  a  longitudinal  direction  or 
parallel  to  the  track.  If  the  structure  is  on  a  curve  it  must  be  made  stable  against  a  lateral 
force,  in  addition  to  the  wind  force,  equal  to  the  centrifugal  force  of  the  train  load.  Transverse 
stability  is  secured  by  giving  to  the  posts  of  the  bent  a  batir,  in  order  to  secure  a  base  of  suffi- 
cient width  to  prevent  overturning.  Longitudinal  stability  is  obtained  by  making  the  span 
which  connects  the  bents  that  are  braced  together  to  form  the  tower  of  sufficient  length.  The 
plane  of  each  bent  should  always  be  vertical.    The  tower  span  is  now  generally  made  30  feet 


392 


MODERN  FRAMED  STRUCTURES, 


long,  and  the  spans  between  the  towers  are  varied  with  the  height  of  the  towers  ;  the  higher 
the  trestle  the  longer  the  intermediate  or  variable  span  may  be  made  with  economy.  The 
approximate  length  for  the  intermediate  span  for  maximum  economy  for  the  height  of  li.e 
trestle  under  consideration  is  usually  determined  upon  first  by  a  rough  calculation,  or  by 
means  of  an  established  formula,  if  the  subject  has  been  previously  investigated.  The 
arrangement  of  spans  and  towers  is  then  made.  It  is  preferable  and  cheaper  to  make  the 
intermediate  spans  of  the  same  length  instead  of  varying  them  slightly  for  variations  in  the 
height  of  the  trestle,  as  it  makes  the  work  simpler  and  gains  a  great  advantage  in  manufacture 
by  having  a  greater  number  of  duplicate  parts.  It  is,  however,  advisable  in  long  viaducts, 
where  much  saving  in  material  can  be  secured,  to  vary  the  intermediate  span  length  for  a 
change  in  height,  if  a  number  of  duplicate  spans  of  each  length  can  be  used. 

The  lengths  of  the  end  spans  of  a  viaduct  are  sometimes  dependent  upon  the  kind  of 
abutments  that  are  built  at  the  ends.  If  T  abutments  are  used  and  the  earth  filled  in 
around  it,  the  first  bent  from  the  abutments  must  be  placed  so  that  the  fill  does  not  cover 
its  base.  If  wing  walls  are  used  to  confine  the  embankment,  the  length  of  the  end  span  may 
be  anything. 

The  outlines  of  the  bracing  are  usually  determined  upon  by  its  appearance  when 
shown  on  a  scale-drawing ;  the  main  object  is  to  avoid  having  the  diagonal  rods  make  too 
acute  an  angle  with  the  vertical.  The  foot  of  each  post  in  a  tower  should  be  braced  by  struts, 
both  longitudinally  and  transversely,  in  order  that  the  tower  may  expand  and  contract  as  a 
whole.  If  the  posts  are  not  so  braced,  the  friction  between  the  base-plate  of  the  column  and 
the  bed-plate  will  produce  a  transverse  stress  and  bending  in,  the  column.  From  this  condi- 
tion  it  can  be  readily  seen  that  the  struts  must  be  made  stiff  enough  to  overcome  the  friction, 
/vnother  benefit  derived  from  the  use  of  these  struts  is  that  the  transverse  force  is  then  resisted 
bj'  boih  the  masonry  pedestals  under  a  bent  to  the  greatest  advantage.  If  the  struts  were 
omitted  the  windward  pedestal  would  have  to  resist  more  than  half  of  the  force,  while  at  the 
same  time  its  vertical  load  would  be  diminished,  and  it  would  therefore  be  less  stable  than  the 
leeward  pedestal,  which  would  have  less  than  half  the  force  to  resist  and  would  have  its  ver- 
tical load  increased. 


Fig.  408. 


In  some  cases  of  low  trestles  it  is  sufificient  to  omit  longitudinal  diagonal  bracing  and 
substitute  knee  braces  or  brackets  between  the  longitudinal  girders  and  the  columns.  The  longi- 
tudinal stiffness  of  such  a  structure  is  measured  by  the  resistance  of  the  columns  to  flexure  in  a 
longitudinal  direction.    A  height  of  from  15  to  20  feet  should  be  the  limit  for  such  designs. 

Rocker  bents,  or  those  hinged  at  both  top  and  bottom,  may  often  be  used  when  it  is  im- 
practicable to  use  a  full-braced  tower.  They  serve  to  shorten  the  length  of  the  spans  and 
brace  the  structure  transversely,  and  on  this  account  are  sometimes  employed  for  economical 
reasons. 


IRON  TRESTLES. 


393 


For  double-track  trestles  the  same  general  arrangement  of  spans  and  towers  is  adhered 
to,  but  the  cross-section  is  varied,  as  shown  in  Fig.  408.  Each  bent  is  composed  of  two,  three, 
or  four  columns  braced  together.  In  designs  {a)  and  (b),  Fig.  408,  the  girders  are  supported 
on  a  cross-girder  at  the  top  of  the  bent.  In  {c)  and  (</)  the  girders  rest  directly  on  the  caps 
of  the  columns.  The  plan  is  a  favorite  way  of  constructing  a  double-track  trestle  from 
an  old  single-track  structure,  or  it  would  be  used  where  it  was  desirable  to  so  construct  a 
single-track  trestle  that  it  may  readily  be  changed  to  double  track.  In  the  latter  case  only 
the  two  inner  girders,  columns,  and  bracing  would  be  built  to  accommodate  a  single  track,  and 
the  outer  girders  and  columns  added  when  provision  is  to  be  made  for  two  tracks. 

A  very  important  difference  in  the  stresses  in  the  bracing  for  single  and  double  track 
bents  is  that  in  the  latter  the  bracing,  if  the  columns  are  battered,  is  stressed  by  the  live  load, 
and  should,  in  order  to  be  consistent  with  the  rest  of  the  design,  be  proportioned  for  lower  unit 
stresses. 

399.  Lateral  Stability  and  the  Batter  of  the  Columns. — It  is  now  generally  specified 
that  the  bents  must  not  require  anchorages  in  order  to  secure  them  against  overturning.  This 
avoids  the  necessity  of  building  large  masonry  pedestals  for  the  columns,  which  would  be 
required  if  an  anchorage  were  needed,  and  further  does  away  with  long  anchor  bolts  and  the 
accurate  measurements  required  to  locate  them  while  building  the  pedestals.  In  order  to 
comply  with  the  condition  that  no  anchorage  must  be  required,  and  also  to  provide  for  the 
excessive  wind  pressures  given  in  some  of  the  standard  specifications,  there  is  no  doubt  that  in 
many  cases  an  excess  of  material  is  used,  and  that  the  total  cost  is  above  that  which  is  actually 
necessary.  All  that  is  actually  necessary  is  that  the  structure  be  made  stable  for  the  highest 
wind  pressure,  which  is  generally  assumed  to  be  50  lbs.  per  square  foot.  The  question  of 
anchorage  should  be  made  one  to  be  settled  by  a  comparison  of  the  total  cost,  including 
pedestals,  with  and  without  anchorage. 

The  batter  of  the  columns  varies  from  \\  to  3  inches  to  the  foot.  If  it  is  desirable  or 
required  that  no  anchorage  be  provided,  it  is  necessary  that  the  wind  and  centrifugal  forces 
combined  with  the  vertical  load  produce  no  resultant  tension  at  the  foot  of  the  windward 
column,  or  at  the  foot  of  the  inner  column  if  on  a  curve.  The  wind  surface  of  a  train  is 
assumed  to  be  10  fget  high,  and  the  centre  of  pressure  7^  feet  above  the  rails.  The  line  of 
action  for  the  centrifugal  force  of  a  train  is  taken  as  5  feet  above  the  rails.  The  amount  of 
the  centrifugal  force  depends  on  the  weight  of  the  train,  the  degree  of  the  curve,  and  the 
velocity  with  which  the  train  is  moving.  For  a  speed  of  thirty  miles  per  hour  the  centrifugal 
force  is  approximately  one  per  cent  of  the  weight  of  the  train  for  each  degree  of  curvature. 
For  speeds  of  forty,  fifty,  and  sixty  miles  per  hour  it  is  approximately  two  per  cent,  three  per 
cent,  and  four  per  cent  respectively  of  the  weight  of  the  train  for  each  degree  of  curvature. 
Thus  for  a  train  weighing  4000  lbs.  per  linear  foot  of  track  moving  at  a  rate  of  forty  miles  per 
hour  on  a  six-degree  curve  the  centrifugal  force  would  be  4000  X  .02  X  6  =  480  lbs.  per  linear 
foot  of  track. 

The  exposed  wind  surface  of  the  structure  is  taken  as  the  surface  of  the  girders  and  floor, 
and  twice  the  surface  of  the  bent  or  tower,  as  seen  in  longitudinal  elevation. 

By  taking  moments  about  the  foot  of  the  leeward  column  the  moment  of  the  vertical 
loads  must  be  equal  to  or  greater  than  that  of  the  wind  forces  if  no  tension  is  allowed  at  the 
foot  of  the  windward  column.  By  making  the  base  of  the  bent  wider,  i.e.,  by  increasing  the 
batter  of  the  columns,  this  can  always  be  accomplished.  By  dividing  the  moment  of  the  wind 
force  by  one  half  of  the  total  vertical  load  on  the  bent  the  distance  which  the  columns  must 
be  spread  apart  at  the  base  is  found  at  once.  The  lightest  train  that  will  not  blow  over  for 
the  assumed  wind  pressure  is  taken  for  the  live  load  on  the  trestle  when  computing  the  width 
of  base.  For  a  30-pGund  wind  pressure  acting  on  a  train  surface  10  feet  high,  beginning  2\ 
feet  above  the  rails,  the  lightest  train  load  is  300  X  7i      2^  =  900  lbs.  per  linear  foot  of  track. 


394 


MODERN  FRAMED  STRUCTURES. 


This  assumption  reduces  the  vertical  load  on  the  column  to  the  minimum  amount,  and  hence 
requires  a  wider  base  for  stability.  The  columns  are  given  the  same  batter  in  all  the  bents  of 
one  structure,  as  it  iniproves  the  appearance  to  have  all  the  columns  on  one  side  in  the  same 
plane,  and  it  also  simplifies  the  manufacture  to  have  all  bevels  alike. 

It  is  often  advantageous,  when  a  viaduct  is  on  a  curve,  to  give  the  outer  columns  of  the 
bents  more  batter  than  the  inner  columns.  This  increases  the  stability  of  the  bent  as  against 
the  combination  of  the  wind  and  centrifugal  forces  for  the  same  width  at  the  base.  The 
bracing  of  the  bents  for  this  case  is  stressed  by  the  vertical  load  because  of  the  different 
inclinations  of  the  columns. 

The  width  centre  to  centre  of  the  girders  or  the  columns  at  the  top  is  variable,  and  is 
independently  determined  for  the  trestle  under  consideration.  The  closer  the  girders  are 
spaced  the  smaller  will  be  the  size  of  the  necessary  cross-ties  and  the  resulting  weight  of 
the  floor,  and  the  bracing  between  the  columns  will  be  shorter  and  will  be  subjected  to  a 
smaller  stress,  requiring  smaller  members,  all  of  which  reduce  the  cost  of  the  structure.  On 
the  other  hand,  the  supported  floor  will  be  very  narrow  and  the  danger  from  a  derailed  train 
increased.  In  case  the  girders  are  spaced  close  centre  to  centre,  it  is  imperative  that  ample 
and  efficient  rerailing  and  safety  appliances  be  used ;  it  is,  however,  advisable  to  use  them  on 
all  trestles.  Another  view  to  take  of  this  question  is  that  generally  taken  by  the  passenger 
departments  of  railroads ;  namely,  that  the  travelling  public  always  feel  much  safer  when  they 
can  see  what  is  supporting  the  train  from  the  car  windows.  This  fancied  security  may  be 
obtained  by  making  the  floor  wider,  and  is  appreciated  by  the  public.  A  good  mean  is  to 
space  the  girders  8  or  9  feet  centres,  as  this  does  not  require  unusually  large  cross-ties  and 
gives  a  safe  floor. 

When  an  old  wooden  trestle  is  to  be  replaced  by  an  iron  structure,  and  traffic  over  the 
trestle  is  to  be  maintained,  it  is  always  economical  to  so  design  the  iron-work  that  it  may  be 
erected  without  disturbing  the  old  bents.  This  is  done  by  spacing  the  girders  far  enough 
apart  to  clear  the  legs  of  the  old  bents  and  locating  the  columns  so  that  they  will  not  be  op- 
posite them.  The  iron  when  erected  will  then  envelop  the  old  work,  and  after  changing  the 
cross-ties  the  latter  can  be  taken  down. 

400.  Length  of  the  Tower  Span  and  Longitudinal  Stability. — It  is  the  common 
practice  to  make  the  tower  span  of  a  viaduct  30  feet  long.  While  this  length  is  generally 
used,  it  is  not  always  true  that  such  towers  do  not  require  anchorage  against  overturning  lon- 
gitudinally. If  it  is  assumed,  as  is  usually  specified,  that  the  friction  on  the  rails  produced  by 
suddenly  applying  the  brakes  to  a  moving  train  is  one  fifth  of  the  vertical  load,  and  further 
that  this  force  is  resisted  hy  all  the  towers  of  the  structure  acting  together,  the  limiting  height 
of  tower  is  about  three  and  one  half  times  its  width  longitudinally  if  no  anchorage  is  allowable. 
The  assumption  that  all  the  towers  act  together  to  prevent  overturning  is  a  fair  one,  as  any 
longitudinal  deflection  of  the  towers  would  result  in  closing  up  expansion  joints  and  thus 
distribute  the  force.  The  coefificient  of  friction  used  is  probably  very  excessive.  In  order 
that  a  series  of  high  towers  may  act  together  to  resist  overturning  with  absolute  certainty,  no 
expansion  joint  need  be  provided  if  the  deflections  of  the  towers  from  temperature  produce 
no  prohibited  stresses. 

401.  Economic  Length  of  the  Intermediate  or  Variable  Span. — No  general  formula 
for  the  length  of  the  variable  span  of  a  viaduct  can  be  given  which  will,  with  the  many  different 
practices  prevalent  among  designing  engineers,  give  reliable  results.  The  length  of  this  span 
for  maximum  economy  depends  upon  the  laws  which  govern  the  variation  of  the  iron  weights 
required  in  the  spans  and  in  the  towers  or  bents,  and  also  on  the  cost  of  the  masonry  pedestals. 
If  the  live  load  and  the  specifications  as  to  allowed  stresses  and  details  of  construction  are 
fixed,  and  also  if  the  designs  are  made  consistent  with  maximum  economy  for  the  various  span 
lengths  and  heights  of  bents,  it  may  be  possible  to  derive  an  empirical  formula  which  will 


IRON  TRESTLES. 


395 


give  results  reliable  enough  to  serve  as  a  guide  in  determining  the  economical  length  of  this 
span.  Such  a  formula  will  be  given,  and  its  origin  and  limitations  explained,  for  the  purpose 
of  enabling  those  who  have  need  of  it  to  derive  one  that  will  suit  their  purposes. 

A  series  of  trestles  varying  in  heiglit  from  25  to  150  feet  were  completely  designed  for  a 
constant  tower  span  of  30  feet,  with  intermediate  spans  of  30,  40,  50,  60,  and  70  feet,  all  plate 
girders.  The  weights  for  each  combination  of  height  of  trestle  and  length  of  intermediate 
span  were  calculated  and  the  results  tabulated. 

A  live  load  of  104-ton  engines  (Cooper's  Class  Extra  Heavy  A)  was  used  in  determining 
tlie  stresses  in  the  girders  and  bents.  The  track  was  straight.  The  wind  pressure  was 
assumed  to  be  30  lbs.  per  square  foot  on  a  train  surface  lO  feet  high,  on  one  surface  of  the 
floor  and  girders  as  seen  in  elevation,  and  on  twice  the  surface  of  the  bent  as  seen  in  elevation. 
The  centre  of  pressure  on  the  train  surface  was  'j^  feet  above  the  rails.  The  track  (rails, 
cross-ties,  etc.)  was  assumed  to  weigh  400  lbs.  per  linear  foot  of  track.  The  permissible  unit 
stresses  were  8000  lbs.  per  square  inch  on  the  net  area  of  the  tension  flanges  of  girders,  8000  lbs. 
per  square  inch  reduced  for  length  by  Rankine's  formula  for  dead  and  live  load  stresses  in  the 
columns  of  the  bents,  12,000  lbs.  per  square  inch  reduced  for  length  by  Rankine's  formula  for 
lateral  struts  and  for  dead,  live,  and  wind  stresses  in  the  columns  of  the  bents,  and  15,000  lbs. 
per  square  inch  in  tension  on  all  bracing  rods  for  the  towers.  The  top  flange  of  all  girders 
was  made  of  the  same  sectional  area  as  the  bottom  flange,  and  the  moment  of  resistance  of 
the  web  was  neglected.  The  longitudinal  bracing  of  the  towers  was  proportioned  to  resist  the 
stresses  resulting  from  a  longitudinal  force  of  800  lbs.  per  linear  foot,  but  the  columns  were 
not  increased  in  area  on  account  of  this  force.  It  is  usually  assumed  that  the  sudden  braking 
of  a  train  on  a  trestle  is  of  rare  occurrence,  and  the  chances  of  it  being  done  for  the  maximum 
train  load  and  during  a  heavy  wind  are  very  remote.  The  thickness  of  the  metal  used  was 
limited  to  three  eighths  of  an  inch  as  a  minimum  everywhere  except  for  bracing,  where  five 
sixteenths  of  an  inch  metal  was  allowed. 

The  plate-girder  spans  were  found  to  vary  in  weight  according  to  the  formula 

w  =  9/-(-  1 10,  (i) 

and  the  weight  of  the  towers  was  very  closely  approximated  by  the  formula 

hi 

w,=  lo.ih  -        •    •    °  (2) 

where  w  =  weight  per  foot  of  the  spans,  /  =:  length  of  span,  =  weight  of  the  towers  per 
foot  of  track,  and  /i  =  height  of  the  towers  from  the  rail  to  the  masonry.  With  these  results 
as  a  basis  the  total  weight  per  foot  of  the  trestle,  IV,  is 

16,200  /il^ 

 +  9A  -430+ 10.3//- ^;  (3) 

where  /,  —  length  of  the  intermediate  span  plus  30  feet,  or  the  distance  between  centres  of 
towers. 

Assuming  that  the  four  pedestals*  under  a  tower  cost  as  much  as  five  thousand  pounds 
of  iron,  the  equivalent  iron  weight  per  foot  of  trestle  which  would  represent  the  cost  of  the 
pedestals  is  5000  ^  Hence  the  total  equivalent  weight  of  iron  per  linear  foot  of  trestle,  in- 
cluding pedestals  is 

16,200  J  ,  5000 


*  The  pedestals  are  assumed  to  cost  the  same  for  each  pedestal,  regardless  of  the  length  of  span,  which  is  generally 

true. 


396  MODEBN  FRAMED  STRUCTURES. 

from  which  W'xsz.  minimum  when 


,  _      /  2I,2CX) 


9-25 


or 

is  a  minimum  for  an  intermediate  span  of  30  feet  when  h  =  77.5 
W"  "       "         "    "  "  "    "  40    "       "    /i=  117.0 

W"  "       "         "    "  "  "    "  50    "       "    k=  142.0 

W"  "       "         "    "  "  "    "  60    "       "      =  159,0 

W"  "       "         "    "         "  '*    "  70   "       "  7^=172.0 

One  very  important  factor  has  been  omitted  in  the  above  investigation,  and  that  is  that 
the  cost  of  manufacture  per  pound  of  the  iron-work  in  the  girder  spans  and  the  bents  differs, 
and  as  the  spans  are  the  cheaper  this  would  tend  to  make  the  economical  span  longer  for  a 
given  height  than  that  found  by  the  formula. 

It  will  be  noticed  that  the  expression  for  the  weight  of  the  towers,  10.3/^  -A,  gives  a 

constant  weight  per  square  foot  of  the  area  included  between  the  rail  and  the  tops  of  the 
masonry  pedestals  for  a  constant  length  of  span,  /,.  Thus  for  /,  =:  60  the  weight  per  square 
foot  of  the  above  area  is  7.9  lbs.,  and  this  decreases  one  twenty-fifth  of  a  pound  for  each  foot 
of  increase  in  the  length  of  /,.  This  gives  an  easily  remembered  expression  for  the  weight  of 
the  towers  and  is  one  very  widely  used.  Rapid  approximate  estimates  of  the  amount  of  iron 
required  in  the  construction  of  trestles  may  be  made  this  way. 

402.  Comparison  of  the  Cost  of  Embankment  and  of  Iron  Trestle. — It  is  often 
desirable  to  know  at  what  height  it  becomes  cheaper  to  use  an  iron  trestle  in  preference  to  an 
embankment.  The  length  of  the  intermediate  span  for  this  case  is  manifestly  the  shortest 
allowable,  or  30  feet,  as  has  become  the  established  practice.  The  weight  per  foot  of  the  girders 
or  spans  would  then  be  the  weight  per  foot  of  30-foot  spans,  and  the  weight  per  square  foot  of 
the  area  of  the  profile  is  that  for  a  length  of  span,  equal  to  60  feet.  If  we  know  the  constants 
to  use  to  get  the  above  weights  and  the  cost  per  pound  of  iron-work  erected  in  place,  the  cost 
of  the  iron-work  per  linear  foot  is  readily  obtained.  The  quantity  of  the  fill  and  the  cost  per 
yard  are  easily  obtained.  Having  the  cost  of  iron  and  of  fill  for  various  heights,  the  height 
at  which  iron  is  the  cheaper  is  readily  found.  This  may  be  reduced  to  a  formula  when  the 
constants  are  determined.  Thus  for  girders  and  towers,  as  in  eq.  (3),  the  weight  of  the 
trestle  would  be  380+  7-9^i  per  linear  foot.  The  number  of  cubic  yards  per  linear  foot  in  a 
/i 

fill  would  be  {d -{-  ^^^)~     The  cost  of  these  two  would  be  equal  when 

/(38o+7-9/^  +  83)  =  (^  +  ^/0^.A.  '  .    .    .  (5) 

where  /  =  cost  in  cents  of  the  iron  per  pound  erected,  /i  =  height  of  the  trestle  or  fill  in  feet, 
l>  —  width  of  the  fill  on  top  in  feet,  s  =  slope  of  the  sides  of  the  fill,  and  =  costs  in  cents  per 
cubic  yard  of  the  fill  in  place.  In  the  above  equation  83  is  the  number  of  pounds  per  linear 
foot  of  trestle  which  is  equivalent  to  the  masonry  pedestals.  If  /  =  4,  />,  =  25,  d  =  14,  and 
5  z=  |.,  we  obtain 

1852      ^i.6k  =  ^14  -j-  "^j ^  =  12,96^  +  i-39^'>    o*"      =  43-8  approximately.* 

*  The  cost  of  the  abutments  is  not  included  in  the  above  formula,  but  an  absolutely  correct  expression  would  take 
them  into  account.  It  would  be  extremely  difficult  to  introduce  this  factor,  as  their  cost  per  foot  of  trestle,  neglecting 
the  question  of  the  kind  of  abutment  used  (i.e.,  T,  U,  or  Wing  abutments),  depends  on  the  length  and  height  of  the 
trestle,  both  being  variable  quantities.  If  it  were  a  question  of  a  fill  throughout,  or  part  fill  and  part  trestle,  such  a 
formula  as  the  one  given  would  not  be  reliable  enough  for  practical  use.  But  if  the  query  were  when  to  stop  the  fill 
and  begin  the  trestle,  the  error  would  be  slight. 


IRON  TRESTLES. 


397 


403.  Design  of  the  Columns. — The  factors  which  should  govern  in  the  selection  of  the 
form  to  use  for  a  trestle  column  are  the  adaptability  of  the  section  for  the  correct  attachment 
of  the  bracing,  and  its  stiffness  and  value  as  a  strut.  The  four-Z-iron  posts  fulfil  the  first 
')ndition  admirably,  but  owing  to  the  small  sizes  of  Z  iron  rolled  it  requires  more  material  in 
|»!C  effective  section  than  is  required  for  the  two-channel  or  two  channel  and  cover-plate  form. 
:  iowever,  as  the  Z  column  requires  no  latticing  and  has  but  two  rows  of  rivets,  it  is  often  the 
ciicaper  section,  and  if  so  is  a  very  desirable  one  to  use. 

While  it  is  not  usually  done,  it  is  no  doubt  advisable  to  make  the  attachments  of  the 
bracing  struts  to  the  columns  very  rigid  in  order  to  add  to  the  stiffness  of  the  column.  For 
high  trestles  it  would  be  advisable  to  reduce  the  allowed  stress  per  square  inch  on  the  columns 
in  order  to  reduce  the  amount  which  they  will  shorten  under  load  and  thus  reduce  the  deflec- 
tion of  the  top  of  the  trestle. 

404..  Lateral  and  Longitudinal  Bracing. — The  bracing  for  the  towers  is  usually  laid 
out  on  a  scale  drawing  so  as  to  appear  to  the  best  advantage  as  bracing  for  the  columns.  The 
heights  of  the  stories  for  the  longitudinal  bracing  is  commonly  made  as  nearly  equal  to  the 
length  of  the  tower  span  as  practicable,  and  in  order  to  brace  the  column  in  both  directions 
at  the  point  of  attachment  of  the  longitudinal  bracing  the  stories  of  the  lateral  or  transverse 
bracing  are  made  the  same  height.  The  attachments  of  all  bracing  should  be  so  made  that 
the  neutral  axes  of  the  members  intersect  on  the  neutral  axis  of  the  column.  Trestle  columns 
are  comparatively  long  struts  and  should  be  as  free  from  eccentric  stress  as  it  is  possible  to 
design  them. 

The  tengio'n  members  of  the  bracing  are  usually  adjustable  rods,  but  of  late  years  angle 
iron  with  riveted  connections  has  been  extensively  used.  The  latter  are  stiffer  but  are  more 
expensive,  and  it  is  extremely  doubtful  if  it  is  possible  to  secure  accurate  adjustment  and  well- 
fitting  riveted  attachments  when  they  are  used. 

The  bottom  flanges  of  the  girders  of  the  tower  span  are  generally  utilized  as  the  top 
longitudinal  strut. 

405.  Stresses  in  the  Towers. — The  external  forces  which  may  act  on  a  trestle  tower 
are  the  wind  pressure  on  the  exposed  surface  of  the  structure  and  the  train,  the  centrifugal 
force  of  the  train  if  the  trestle  is  on  a  curve,  the  friction  of  the  wheels  on  the  rail,  the 
weight  of  the  structure  and  the  train,  and  the  reactions  of  the  pedestals.  The  wind,  the 
centrifugal  force,  and  the  vertical  load  if  not  applied  centrally,  or  if  the  bent  is  not  symmetrical, 
stress  the  transverse  bracing.  The  friction  of  the  wheels  alone  puts  a  stress  on  the  longi- 
tudinal bracing.  JThe  bottom  struts  of  both  the  longitudinal  and  transverse  bracing  must  be 
made  strong  enou|;h  to  resist  a  stress  equal  to  the  friction  of  the  column  base  on  the  bed- 
plate in  order  to  provide  for  the  effects  of  temperature.  The  amount  of  this  friction  may  be 
assumed  for  safety  as  twenty-five  per  cent  of  the  dead  load  on  the  bed-plate. 

The  stresses  ra  the  towers  may  be  found  by  the  methods  given  in  Part  L  The  stresses 
for  all  the  external  forces  should  be  computed,  as  it  is  a  very  common  mistake  to  overlook  the 
effect  of  the  vertical  load  on  the  bracing. 

The  stresses  in  the  towers  are  usually  found  analytically  as  follows:  Referring  to  Fig.  409,  which  is  the 
general  case  of  a  single  track  bent  with  columns  of  different  batters,  the  stresses  for  the  loading  shown 
would  be : 

Maximum  stress  m  AB  =  \  WM  +  >J)  +  W.,{b     k)  ^  IF.ic  +  Ji:)+  lVvj\  -i- {c -if- k) ; 
"  CB  =^\W,a-\-  W.,b-ir  IV^c^  lV^x\^y; 
"  CD=  \  lV,a  -f  IV^b  +  W,c  +  lV,{c  -f  d)  +  IV^x  [  -s-  (<-  +  </)  ; 
"  ED=\  W,a  4-  lV.,b  -f  IV^c  -I-  W^(c  -\- d)  +  W^x  \^z: 
"AD  =  \  W^'a^  +  W^'b  +  W^U  -  W^x  \  -^y, ,    or    ]  PV^'b  -|-  W,'c  \-*-yx\ 


t€  << 

;.- 

((  « 
«•  it 


398 


MODERN  FRAMED  STRUCTURES. 


Maximum  stress  \n  CF=\  Wy'ay  +  IV^'b  +  IV^'c  +  IV^'ic -\- d)  -  W^x  [  -i-  2, ,    or    ]  W^'b  +  W^'c  +  IV^'ic  +  d)  \ 
"      '•  BD  =  \w,{e-\-d-a)+  lV^{c  +  d-b)-\-  lV^d+  W^f\-^w; 

"     ••  £>F=\lV,{c+d+^-n)+  W,{c+d+e-6)+  lV,(d+e)+  lV,c+lV^^\-i-u; 
••  "      '•  ^C=\  lV,'{c  +  d- a,)+  lV.,'(c  + d  -  b)+  lV,'d+  Jr.Ji\^  wr, 

CE  =  \  lVi\c  ^d+e-ai)+  IV^'ic-  +  d+e-b)+  W^'id+e)  +  W^'e  +  IV J  [  + 


E 

F 

Fig.  409. 

The  maximum  stresses  given  above  for  BD,  DF,  AC,  and  CE  are  the  maximum  compressive  stresses 
only.  The  two  expressions  for  .4Z>anfi  CF  mean  that  these  members  may  possibly  receive  their  maximum 
stresses  when  the  train  is  not  on  the  trestle.  This  would  be  the  case  if  W-Jx  were  greater  ihan  Wi'ax ,  as 
these  two  expressions  include  all  the  forces  due  to  the  train. 


IRON  TRESTLES. 


399 


In  the  above  expressions  W.„  is  that  part  of  the  weight  of  the  train,  track,  and  girders*  which  is 
supported  by  the  bent,  and  the  force  is  assumed  to  act  vertically  through  the  centre  of  the  track  if 
the  rails  are  of  the  same  elevation,  or  as  many  inches  from  the  centre  of  the  track  towards  the  low 
rail  as  the  outer  rail  on  a  curve  is  elevated  above  the  inner  rail,  if  the  rails  have  different  elevations. 
Wi  and  W\'  are  the  combined  effects  of  the  wind  pressure  on  the  train  and  the  centrifugal  force. 
The  wind  pressure  on  a  train  is  usually  taken  as  300  lbs.  per  linear  foot  of  track,  acting  ^\  feet  above 
the  rails.  The  centrifugal  force  is  assumed  to  act  5  feet  above  the  rails,  ffj  and  H^j'  represent  the 
wind  pressure  on  the  girders  and  floor,  generally  taken  as  30  lbs.  per  square  foot  if  the  train  is  on  the  struc- 
ture, and  50  lbs.  per  square  foot  if  the  structure  is  not  loaded.  W% ,  W»',  and  14^*'  represent  the  wind 
pressure  on  the  tower  concentrated  at  the  joints.  This  pressure  on  the  bents  is  generally  assumed  as  225 
lbs.  per  foot  of  height  of  bent  for  the  loaded  structure,  and  375  lbs.  for  the  structure  not  loaded.    Thus  IVa 

d  d 

and  Wi  would  be  taken  as  225  .  — and  375.— ,  for  the  loaded  and  unloaded  structure  respectively,  and 

d  Jr  e  d  +  e 

Wi,  Wt  would  be  225  .        and  375  .  — ^  similarly. 

Bents  with  the  columns  inclined  differently,  such  as  the  one  illustrated,  are  seldom  used  except  for 
trestles  on  a  curve,  and  the  example  is  given  mainly  to  show  the  effect  of  the  load  on  the  bracing.  For 
double  track  bents  with  inclined  columns  a  train  on  one  track  only  will  always  stress  the  bracing. 

The  bracing  is  proportioned  to  resist  the  stresses  due  to  all  of  the  above  external  forces 
except  that  of  the  vertical  train  load  at  the  limiting  allowed  unit  stresses  used  for  wind 
bracing  generally.  For  stresses  resulting  from  the  vertical  train  load  the  limiting  stresses  are 
those  used  in  the  counter-ties  and  posts  of  trusses.  For  the  stresses  in  the  columns  from 
wind,  centrifugal  force,  and  the  friction  of  the  wheels  on  the  rails,  the  practice  varies.  Some 
specifications  require  that  the  area  of  the  column  be  increased  for  these  stresses  beyond  that 
required  by  the  vertical  load  by  an  amount  in  square  inches  equal  to  the  stress  divided  by  the 
value  of  the  column  per  square  inch  as  a  lateral  or  wind  strut.  Other  specifications  require 
no  increase  in  section  until  the  stresses  from  the  above  forces  increase  the  stress  per  square 
inch  fifty  per  cent  above  that  allowed  for  the  vertical  load  on  the  column  and  then  add 
section  until  the  total  stress  per  square  inch  from  dead  load,  live  load,  wind  and  centrifugal 
force  does  not  exceed  that  amount.  It  is  generally  assumed  that  the  wind  force  and  that  of 
the  friction  of  the  wheels  on  the  rails  are  not  coexistent  forces.  The  rational  way  to  propor- 
tion the  columns  would  seem  to  be  to  use  low  permissible  stresses  for  the  working  or  duty 
stresses,  which  are  those  due  to  the  dead  and  live  loads,  including  centrifugal  force,  and  to 
increase  the  permissible  stresses  as  much  as  is  consistent  with  safety  for  wind  stresses  in  com- 
bination with  the  dead  and  live  load  stresses.  In  the  case  of  a  trestle  near  a  switch  or  a 
station  where  trains  are  continually  using  air-brakes,  it  would  be  advisable  to  include  friction 
stresses  among  the  duty  stresses. 

406.  The  Masonry  Pedestals  supporting  a  tower  are  usually  made  as  small  as  it  is 
possible  to  make  them  and  not  exceed  the  permissible  pressure  on  the  foundation.  They 
should  be  made  as  low  as  possible,  a  good  rule  being  to  have  them  extend  one  foot  above  the 
surface  of  the  ground,  in  order  to  make  them  stable  against  the  lateral  pressure  of  the 
wind.  In  all  cases  their  stability  should  be  investigated  and  ample  provision  made  for  the 
extreme  case. 

407.  Details  of  the  Towers. — Figs.  410,  411,  and  412  show  the  ordinary  details  for  a 
tower  with  Z-columns.  The  struts  are  all  riveted  to  the  columns,  and  the  rods  have  pin  con- 
nections. The  intersections  are  as  near  the  neutral  axis  of  the  column  as  it  is  practicable  to 
make  them.  If  the  rods  are  very  large,  or  if  they  are  stressed  by  the  live  load,  it  is  impera- 
tive that  the  intersections  of  the  neutral  axis  of  the  struts  and  rods  at  a  joint  be  on  the 

neutral  axis  of  the  column  ;  but  for  the  usual  case  of  small  rods  a  slight  eccentricity  is  usually 

 -J  . — 

*  An  absolutely  correct  expression  would  separate  Wv  into  two  parts,  that  due  to  the  train  load  and  that  of  the 
track  and  girders.  It  is  not  done  here,  as  it  would  complicate  the  illustration,  and  as  it  is  only  necessary  in  the  unusual 
case  of  a  bent  having  columns  of  different  batters. 


40O 


MODERN  FRAMED  STRUCTURES. 


permitted.  If  posts  composed  of  two  channels  are  used,  the  transverse  bracing  may  be  con- 
nected by  pins  through  the  neutral  axis  of  the  column.  The  details  shown  will  serve  to  show 
one  method  of  connecting  the  bracing,  but  any  other  which  accomplishes  the  same  results 
would  be  satisfactory. 

The  Caps  of  the  Columns. — At  this  point  provision  must  be  made  for  supporting  the 
girders  and  for  attaching  the  transverse  and  longitudinal  bracing.  The  girder  rests  on  a 
cap  plate  riveted  to  the  top  of  the  column  by  means  of  auxiliary  angles  as  shown.  The  top  of 
the  column  is  planed  to  an  even  surface,  on  which  the  cap  plate  bears  and  transfers  the  load 
of  the  girder  directly  through  the  bearing.  The  attachment  of  the  transverse  bracing  is 
usually  simple,  the  strut  and  the  column  resisting  the  two  components  of  the  stress  in  the  rod. 
In  the  case  shown,  Fig.  410,  a  bracket  is  riveted  to  the  column,  to  which  is  riveted  the  strut, 


Fig.  410. 


and  the  rods  are  connected  to  it  with  a  pin.  The  attachment  of  the  top  transverse  strut  must 
be  made  sufficient  to  take  the  horizontal  component  of  the  stress  in  the  column.  This  com- 
ponent is  best  transferred  to  the  strut  by  rivets  through  the  cap  plate  extended  as  in  the 
sketch.    The  top  strut  is  usually  short,  and  can  be  made  cheaply  of  two  angles. 

Intermediate  Connections  to  the  Column. — At  the  intermediate  joints  the  attachments  of 
the  bracing  are  very  simple,  as  the  only  requirement  is  that  the  rivets  through  the  column  and 
through  the  struts  be  sufificient  to  transfer  the  respective  components  of  the  stresses.  If  the 
rods  are  small,  a  slight  eccentricity  in  these  attachments  is  allowable. 

A  splice  in  the  column  should  be  made  a  short  distance  above  the  joint  by  planing  the 
ends  of  the  two  sections  for  an  even  bearing  and  riveting  splice  plates  on  all  four  sides  to 
hold  the  sections  in  line,  but  relying  on  the  butt-joint  for  the  transfer  of  the  stress. 

408.  Column  Bases. — At  the  foot  of  the  column  the  bracing  is  attached  as  before,  but 
here  provision  must  be  made  for  the  distribution  of  the  pressure  over  the  masonry,  so  as  not 
to  exceed  the  permissible  bearing  pressure.  The  column  is  planed  to  an  even  surface  and 
transfers  the  load  directly  to  the  sole-plate,  which  is  stiffened  by  gusset  plates  and  angles 
riveted  to  it  and  to  the  column.  The  sole-plate  at  an  expansion  point  should  rest  on  a  bed 
plate,  in  order  that  the  friction  or  resistance  to  the  movement  of  the  column  at  this  point  may 
be  as  little  as  possible.  The  bed-plate  must  be  made  large  enough  and  sufficiently  stiff  to 
distribute  the  pressure  without  exceeding  the  limiting  bearing  pressure  per  square  inch. 

The  holes  for  the  anchor  bolts  should  be  made  circular  in  the  bed-plate  and  oblong  or 
slotted  in  the  sole-plate,  to  allow  the  latter  to  slide  on  the  former. 


IRON  TRESTLES. 


MODERN  FRAMED  STRUCTURES. 


III.  ELEVATED  RAILROADS, 

409.  Characteristic  Features. — -An  elevated  railroad  differs  from  a  trestle  in  being 
lower,  and  in  the  fact  that  it  usually  occupies  the  streets  of  a  city.  Tlie  height  is  the  least 
allowable  for  wagon  traffic  beneath  it,  and  the  provision  for  this  traffic  usually  prohibits  the 
use  of  vertical  diagonal  bracing,  either  longitudinal  or  transverse.  It  is  therefore  a  railway,  of 
two  or  more  tracks,  supported  upon  a  series  of  short  iron  or  steel  columns,  providing  from 
fourteen  to  sixteen  feet  clearance,  without  complete  systems  of  vertical  diagonal  bracing. 
Such  railroads  generally  have  a  very  heavy  traffic  also,  especially  in  the  number  of  the  trains; 
and  this  demands  a  very  rigid  structure,  not  liable  to  be  racked  to  pieces.  Because  the 
diagonal  vertical  bracing  cannot  extend  to  the  foundations,  there  may  be  very  large  bending 
moments  produced  in  the  columns,  from  lateral  wind  and  centrifugal  forces,  and  from  the 
rapid  slowing  up  of  trains  when  stopping,  which  produce  great  longitudinal  thrusts.  If 
the  columns  are  free  to  turn  at  either  end,  as  they  would  be  if  pin-connected  or  insecurely 
attached  at  these  points,  the  bending  moment  is  about  twice  as  great  in  them  as  it  is  when 
they  are  rigidly  held  by  adequate  anchorage  and  top  connections.  It  is  very  important, 
therefore,  to  provide  such  anchorage  and  connections  as  will  effectually  fix  the  posts  in  direc- 
tion at  their  extremities,  this  being  one  of  the  most  important  features  of  this  class  of  structures. 

As  to  styles  of  superstructure,  it  may  be  said  that  both  the  girders  and  floor-beams  are 
now  made  up  with  solid  webs,  or  as  plate  girders.  If  a  lighter  structure  is  desired  than  can 
be  obtained  in  this  way,  then  a  riveted  lattice  girder  may  be  used  ;  but  great  care  should  be 
given  to  its  design  and  construction,  a  sufficient  number  of  rivets  used  in  making  the  joints, 
and  all  gravity  lines  made  to  intersect  in  a  point  at  every  joint  if  possible.  Both  legs  of  all 
web  angles,  also,  should  be  joined  to  the  chord  sections,  so  that  these  members  can  receive 
their  loads  over  their  entire  cross-sections  and  not  through  one  leg  only,  as  is  so  often  done. 
As  for  pin-connected  trusses  for  such  structures,  they  should  not  even  be  considered. 

In  the  following  articles  on  this  subject  only  such  matters  will  be  discussed  as  are  peculiar 
to  this  class  of  structures,  and  a  few  of  the  best  examples  illustrated. 

It  goes  without  saying  that  an  all  masonry  or  embankment  support  for  an  elevated  rail- 
way (except  on  street-crossings)  is  far  superior  to  one  of  steel  girders  and  columns  ;  but  when 
the  road  occupies  the  public  streets  or  alleys  this  is  impracticable,  and  in  any  case  it  is  more 
expensive.  The  elevated  railway  system  of  Berlin  is  built  in  this  way,  largely  for^the  pur- 
pose of  preventing  the  noise  which  always  accompanies  a  rapidly  moving  train  upon  a 
metallic  structure.  The  tracks  of  the  Pennsylvania  Railroad  Company  in  Philadelphia  are 
also  carried  upon  a  masonry  structure.  On  the  Berlin  system  the  tracks  on  the  bridges  which 
cross  the  streets  are  embedded  in  gravel,  also  for  the  purpose  of  preventing  the  noise  of 
passing  trains. 

The  details  of  an  elevated  structure  should  be  worked  out  with  great  care,  with  a  view  to 
permanent  stability  as  well  as  to  economy.  The  duplication  of  parts  causes  small  savings  in 
design  to  accumulate  to  considerable  sums  in  many  miles  of  the  structure. 

410.  The  Live  Loads. — Elevated  roads  built  for  standard  traffic  are  of  course  designed 
to  carry  the  same  engine  and  train  loads  as  the  bridges  of  the  same  line.  Roads  built  for  city 
or  suburban  passenger  rapid-transit  service  are  designed  for  the  particular  styles  of  motors  and 
cars  which  are  expected  to  run  upon  them.  It  has  been  common,  however,  in  this  class  of 
structures  to  greatly  underrate  the  future  requirements,  so  that  within  a  few  years  of  the 
inauguration  of  the  system  the  actual  loads  are  far  in  excess  of  those  for  which  the  structure 
was  built. 

The  following  wheel-load  diagram,  Fig.  413,  which  gives  the  loads  on  one  rail,  or  one 
half  the  train  loads,  was  used  in  designing  the  Chicago  and  South  Side  Elevated  in  1890. 


ELEVATED  RAILROADS.  403 

This  provides  for  forty-four-ton  engines  and  thirty-two-ton  cars,  which  are  both  considerably 
greater  than  are  actually  used. 

Oiijine  >^  Car  —  —  —      —  —  —  «  _   _  „  ^  Car^  >e  - 

,6:41 ,4:65   ,      ,  ,  ... 


Q  n  O  n  O  O  on       DO  o  o  Q_ 


i  i  i  i 

;g  ^ 


Fig.  413. 


It  has  been  customary  to  provide  for  thirty-ton  *  engines  and  twenty-eight-ton  cars,  hav- 
ing a  wheel  base  as  shown  in  Fig.  414,  where  the  loads  on  one  rail  are  again  shown. 


Fig.  414. 


The  computations  of  stresses,  and  the  dimensioning  of  all  the  parts,  have  been  fully 
explained  elsewhere  in  this  work. 

411.  The  Lateral  and  Longitudinal  Stability  and  stiffness  of  these  structures  is 
dependent  upon  the  stiffness  of  the  columns.  Three  different  methods  have  been  used,  in 
which  the  object  sought  was  to  secure  the  greatest  stiffness  with  the  least  cost.  The  old 
method  was  to  anchor  the  columns  firmly  to  adequate  foundations,  and  connect  the  cross  and 
longitudinal  girders  to  them  loosely  at  the  top.  In  this  plan  the  column  was  assumed  to  be 
anchored  at  the  bottom  and  "  free  ended  "  at  the  top.  Tlie  bending  moment  which  the 
column  was  proportioned  to  resist  was  equal  to  the  horizontal  force  at  the  top  multiplied  by 
the  height  of  the  column.  Another  method  is  the  exact  reverse  of  the  above,  as  in  this  case 
the  connection  of  the  cross  and  longitudinal  girders  with  the  column  is  made  rigid,  and  the 
connection  with  the  foundation  is  made  only  sufficient  to  resist  lateral  displacement.  The 
columns  in  tliis  case  are  assumed  to  be  fixed  or  anchored  at  the  top  and  "  free-ended  "  at  the 
bottom.  The  bending  moment  for  which  the  column  is  to  be  proportioned  is  the  same  as  in 
the  forrr^r  method.  The  latest  method  is  a  combination  of  these  two,  as  in  this  plan  the 
columns  are  anchored  rigidly  to  adequate  foundations  at  the  bottom,  and  to  the  cross  and 
longitudinal  girders  at  the  top.  Changes  of  temperature  produce  a  small  stress  in  the  structure 
when  this  plan  is  used.  It  is  probably  insignificant,  but  should  always  be  computed.  The 
columns  are  therefore  assumed  to  be  fixed  in  direction  at  both  ends,  and  the  bending  moment 
which  they  are  proportioned  to  resist  is  about  one  half  of  the  product  of  the  horizontal  force 
into  the  height  of  the  column.  The  first  and  third  methods  require  expensive  and  reliable 
foundations,  while  the  second  method  requires  the  least  amount  of  foundation  possible  in  any 
case.  The  first  and  second  plans  require  about  the  same  amount  of  iron  in  their  construction, 
and  more  than  is  required  by  the  third  plan. 

If  the  design  of  the  structure  is  such  that  each  track  is  supported  by  a  single  line  of 
columns  that  have  no  connection  with  the  columns  under  the  other  track,  it  is  absolutely 
necessary  to  "fix"  or  anchor  the  columns  to  the  foundations,  as  this  is  the  only  means  of 
securing  lateral  stability. 

Double-track  structures  have  been  supported  on  a  single  line  of  columns,  and  in  such 
cases  the  anchorages  must  be  able  to  resist  the  bending  moment  due  to  the  eccentric  loading 
of  the  tracks  in  addition  to  that  of  the  lateral  forces. 


*  Thirty-five-ion  engines  have  been  used  on  some  of  the  elevated  passenger  roads  in  New  York  City, 


404 


MODERN  FRAMED  STRUCTURES. 


The  wind  pressure  on  elevated  railway  structures  is  taken  at  30  lbs.  per  square  foot  of 
exposed  surface,  and  the  protection  afforded  by  adjacent  buildings  is  usually  neglected.  The 
longitudinal  force  is  assumed  to  be  that  due  to  the  sudden  application  of  air-brakes  to  a 
moving  train,  checking  the  wheels  and  causing  them  to  slide  on  the  rails.  The  coefficient  of 
friction  is  taken  as  \. 

Expansion  Joints  are  provided  in  elevated  railway  structures  at  intervals  of  about  200 
feet.  For  riveted  trusses  the  longitudinal  girders  slide  on  the  top  flange  of  the  cross-girder,  as 
shown  in  Fig.  420.  The  usual  expansion  joint  for  plate-girder  construction  is  shown  in  Plate 
XI.  The  longitudinal  girder  rests  freely  on  a  shelf  or  bracket  built  out  from  the  cross-girder. 
It  is  customary  to  have  the  longitudinal  girders  on  one  side  of  the  cross-girder  rigidly  connected 
to  the  cross-girder  at  an  expansion  joint. 

When  the  expansion  joint  is  a  suspended  one,  as  when  the  girder  and  cross-beam  have 
about  the  same  depth,  the  construction  shown  in  Figs.  415  and  416  may  be  followed.* 


k  /5<|  *  


coff/./s'if'/e'ai' 


Fig.  415. 


1  1 

i  i 
i  i 

_J 

Lit 

1  1 

1  1 
t  > 

1  r1- 

1  !  I 

}&- 

-0- 

i 

1  1  1 

_}._.]  1 

i 

( 

!  ! 

1  1 

1 

Section  A-B. 

Fig.  416. 


The  merit  of  this  joint  lies  in  the  cast-iron  filler  piece  having  a  curved  bottom  which  fits 
the  curve  given  to  the  pocket-plate.    By  the  aid  of  this  cast-iron  support  the  upward  pull  on 


*  These  are  from  plans  prepared  for  the  Quaker  City  and  Northeastern  Elevated  Railways  of  Philadelphia.  See 
Engineering  News,  May  25,  1893. 


ELEVATED  RAILROADS. 


the  sides  of  the  pocket-plate  is  transmitted  uniformly  over  the  bearing  area  of  the  longitu- 
dinal girder. 

The  casting  is  bolted  to  the  pocket-plate,  and  the  girder  slides  upon  it. 

Foundation  Joints. — An  accurate  and  even  bearing  of  the  columns  upon  the  cap  stones  of 
the  foundations  is  of  the  utmost  consequence.  It  is  impossible  to  make  these  parts  fit  exactly 
after  the  structure  is  riveted  up,  however  much  care  may  have  been  taken  in  the  shop  and 
foundation  work.  If  the  bearing  is  not  even,  the  weight  of  the  structure  with  its  load  will 
come  upon  one  side  of  the  column  section,  thus  giving  to  it  an  excessive  bending  moment 
which  it  was  not  designed  to  carry,  and  greatly  overstressing  a  portion  of  the  cross-section. 

To  insure  an  even  bearing  it  is  absolutely  necessary  to  wedge  up  the  post,  after  riveting 
up,  until  it  bears  evenly  on  all  sides,  and  until  it  is  raised  about  three  eighths  of  an  inch  from 
the  stone.  Then  fill  this  space  with  Portland  cement  or  iron-rust  cement,  made  of  iron  filings 
or  trimmings  mixed  with  sal-ammoniac.  This  should  be  rammed  to  place  with  a  thin  steel  or 
hard  wooden  blade,  and  allowed  to  harden  before  the  wedges  are  removed,  and  the  anchor-bolt 
nuts  screwed  down  hard.  If  the  posts  are  socketed  in  cast-iron  pedestals  and  sealed  in  place  by 
long  iron-rust  joints,  as  described  in  Art.  413,  the  ideal  conditions  are  probably  realized. 

412.  General  Case  of  Horizontal  Forces  Acting  on  an  Elevated  Railway  with  Columns  fixed  at  the 
Ground.— In  Fig.  417  let  the  horizontal  force  P  act  u[)oii  the 
train  and  structure,  so  that  its  centre  of  pressure  is  at  the 
height  h  from  A  and  h'  from  A' ,  the  supports  being  on  ground 
not  level  transversely.  Let  the  lateral  system  CDD'C  be  sup-  R-- 
posed  to  be  rigid  as  compared  to  deflections  of  the  columns,  so 
that  the  conditions  assumed  in  Art.  151,  Chap.  X.  are  fulfilled. 
Then  we  have,  from  eq.  (20),  Art.  151, 


z 

Xo  —  — 
2 


Z  +  2C 
2Z  +  C, 


2  \2z'  +  C  J 


(6) 


To  find  the  relative  values  of  //and thesum  being  equal 
to  P,  we  may  resort  to  the  use  of  Proposition  III,  Art.  200,  Chap. 
XV,  according  to  which  the  load  P  divides  itself  between  the 
columns  in  direct  proportion  to  their  relative  rigidities;  or  in- 
versely as  their  deflections  at  the  points  P>  and  £>',  for  the  same 
horizontal  forces,  or  shears,  acting  between  A  and  B  in  the  one 
case  and  between  A'  and  D'  in  the  other.  Hence  we  may 
write 


H' 


J' 


(7) 


where  A  and  A'  are  unequal  horizontal  deflections  of  D  and  D' 
respectively  for  the  same  horizontal  reaction  acting  throughout 
their  lengths  from  A  and  A'  to  D  and  D'. 

For  this  condition  it  was  shown  in  Art.  151,  eq.  (20),  that 
the  point  of  inflection  is  situated  at  a  distance  above  the  base  equal  to  (Fig.  417) 


Fig.  417. 


Xa  = 


Z    Z  +  1C 


2\2Z  +  C 


3^  +  2e\ 
32 +  e)- 


(8) 


Since  the  column  may  be  supposed  to  be  cut  at  this  point  of  contraflexure  the  given  horizontal  reaction.  A', 
may  be  supposed  to  act  here  towards  the  left  on  the  lower  portion  (Fig.  417)  and  towards  the  right  on  the 
upper  portion  of  the  column,  or  the  moment  at  any  part  of  the  column  below  L>  may  be  expressed  by  the 
equation 

Mx<:=  K(Xa  —  X)  (9) 

To  find  the  deflection  of  the  point  D  with  reference  to  A,  we  have  from  the  equation  of  the  elastic  line 

M  K 

(10) 


cPy  _  _ 

dx^-EI=EI^'''~''^- 


*  In  this  equation  c  —  k -\-  e  oi  Fig.  417. 


4o6 


MODERN  FRAMED  STRUCTURES. 


Integrating, 


since  for  x  =  o,  =  o. 


dx  ~  EI 


7(+c  =  ojJ.   .  (II) 


^=^L~-^l+^=°)J  


since  for  x  —  o,  y  —  o. 

For  X  =  z,  vie.  have_y  =  A,  the  deflection  at  I),  or 


EI 


XtsZ' 
2 


and  similarly  for  the  deflection  at  D'  for  the  same  force  K,  in  the  other  column, 


~  Ef'[_  2    ~  6  . 


(13) 


(14) 


From  eq.  (8)  we  have 

and  similarly. 


Xa  = 


c\      z  (3z+2e\ 

c)='2[^^Tr)' i   .    .    .  (15) 


2  \2z'  +  e 

where  e  =  c  —  z,  -a^  shown  in  Fig.  2o8rt. 

Substituting  these  values  of  jto  and  xv  in  eqs.  (13)  and  (14),  we  obtain 


A  = 


z^K   I   Y2,z  +  2e  ' 


12EI 


and  similarly. 


Whence,  from  eq.  (7),  we  obtain 


A'  = 


z'^K 

~i2EI'\ 


3  +  4p' 
3  + 


But  H  +  H'  =  P,  therefore. 


H  A'  z'^I 
W  ~  ^  ~¥l' 


H' 


e 

3  +  4- 

z' 


3  +  4: 


3  + 


e 

3  +  3 


z'U 

zW\  r 


and 


e 

3  +  4- 


3  + 


e 

3  +  4- 
z 


e 

3  +  ? 


^  +1 


H  =  P-  H'. 


If  I  =  /',  both  of  these  disappear  from  the  formulae. 

If  z  =  z',  or  if  the  supports  are  on  a  level,  then  eq.  (20)  becomes 

PI' 


H'  = 


/+/" 


P 

=  -    when  /  =  /', 


(16) 


z^K  j 

I'-  • 

\2EI  \ 

+  F  V 

(17) 


(18) 


(19) 


(20) 


(2U 


(22) 


ELEVATED  RAILROADS.  407 

If  e  =0,  which  corresponds  to  the  case  of  a  column  fixed  in  direction  at  D,  as  when  the  columns  are 
joined  by  a  plate  girder  extending  from  C  to  D,  we  have,  from  eq.  (20), 

To  find  the  point  of  application  of  H,  or  the  point  of  inflection,  we  have  from  eqs.  (15)  and  (16): 
For  z  =  s' ,  or  for  supports  on  a  level, 

For  e  =  o,or  for  columns  fixed  in  direction  at  D,  as  by  a  plate  girder  from  D  to  C,  we  have 

•*^o  =  -;  (25) 


or  the  point  of  inflection  is  midway  between  A  and  D. 

For  e  =  z,  or  for  open  bracing  in  the  upper  half  of  the  bent,  we  find 

x„  =  ^z  (26) 

or  the  point  of  inflection  is  f  of  the  distance  AD  from  A. 

After  having  found  H  and  H'  and  their  points  of  application  for  any  case,  the  resulting  bending 
moments  and  direct  stresses  are  readily  computed  by  the  ordinary  methods  of  Art.  115. 

413.  Selected  Examples. — In  Plate  XI  arc  shown  the  general  and  detail  drawings  of 
the  elevated  portion  of  the  Merchants'  Terminal  Raihvay  of  St.  Louis.  It  is  of  the  plate- 
girder  style  of  construction  throughout,  and  carries  a  double  track  for  standard  railway  trafific. 
It  is  provided  with  an  expansion  joint  in  every  second  panel,  and  the  posts  are  effectually 
anchored  to  the  ground.  The  bearings  on  the  foundation-stones  were  made  by  packing  iron 
filings  and  sal-ammoniac  between  foot-plate  and  cap-stone  after  the  structure  was  riveted  up. 
The  columns  are  assumed  to  be  fixed  in  direction  at  both  top  and  bottom.  Wherever  it  was 
possible  both  the  lateral  and  the  longitudinal  bracing  reached  to  the  bases  of  the  columns. 
Where  this  was  not  possible  an  auxiliary  lattice  longitudinal  bracing  was  used  where  the 
plate  stringer  was  higher  than  14  feet  from  the  pavement,  as  shown  in  the  plate. 

This  structure  was  built  by  the  Phoenix  Bridge  Company,  under  the  direction  of  Robert 
Moore,  M.  Am.  Soc.  C.  E.,  as  Chief  Engineer,  and  is  thought  to  be  one  of  the  best  examples 
of  standard  elevated  railway  construction. 

The  Chicago  and  South  Side  Elevated  Railway  is  about  eight  miles  long,  and  extends 
from  the  heart  of  Chicago  to  Jackson  Park.  It  is  built  mainly  on  a  purchased  right  of  way 
30  feet  wide,  alongside  an  alley.  It  is  designed  for  passenger  trafific  only.  The  general 
plans  are  shown  in  Figs.  418  and  419.  The  stringers  are  plate-girders  forming  spans  of  from 
35  to  60  feet.  The  standard  structure  has  no  cross-beam,  properly  speaking,  the  girders  being 
carried  directly  by  the  columns,  the  tops  of  which  spread  sufificiently  to  do  this.  In  place  of 
the  cross-beam  is  a  system  of  diagonal  bracing,  as  shown  in  Fig.  419.*  The  columns  are 
firmly  anchored  to  a  foundation  of  masonry  7  feet  square  at  the  base  and  reaching  to  a  depth 
of  10  feet.  The  upper  portion  of  this  is  a  cast-iron  cap  24  inches  high,  provided  with  sockets 
2\\  in.  deep  to  receive  the  channel  bars  forming  the  columns.  This  casting  is  bolted  to  the 
masonry  by  four  \\-m.  anchor  rods,  and  the  channel  bars  are  absolutely  joined  to  the  cast-iron 
base  by  means  of  an  iron-rust  cement  composed  of  iron-filings  and  sal-ammoniac,  which  was 
tamped  into  the  surrounding  spaces  after  the  structure  was  riveted  up.  Mr.  Robert  S.  Sloan, 
M.  Am.  Soc.  C.  E.,  was  Chief  Engineer,  and  it  was  erected  by  the  Keystone  Bridge  Com- 
pany of  Pittsburg. 


*  For  a  detailed  and  illustrated  description  of  this  structure,  with  its  passenger  stations,  see  Eng.  News  of  Jan.  16, 
1892. 


MODERN  FRAMED  STRUCT  URES. 


ELEVATED  RAILROADS,  409 


MODERN  FRAMED  STRUCTURES. 


The  Kings  County  Elevated  Passenger  Railway  of  Brooklyn,  New  York,  is  illustrated  in 
Fig.  420.  This  structure  was  designed  in  1885  and  built  in  1887.  The  charter  required  it  to 
be  of  the  lattice-girder  type,  and  to  be  placed  over  the  centre  of  the  street.  It  was  designed 
and  built  by  the  Phoenix  Bridge  Company.  The  cross-girders  are  in  pairs,  resting  on  Phcenix 
columns,  being  joined  only  through  the  top  plate  of  the  column.  The  structure  is  cheap 
in  first  cost,  but  not  as  rigid  as  it  should  be  for  heavy  traflfic.  The  columns  are  not  very 
rigidly  attached  at  either  top  or  bottom,  and  as  for  longitudinal  stability,  it  has  very  little. 
It  is  by  no  means  an  ideal  structure,  but  a  fair  example  of  the  earlier  forms  of  rapid-transii 
elevated  railways. 

Plate  XII  contains  the  general  drawings  of  the  four-track  steel  viaduct  of  the  New  York 
Central  and  Hudson  River  Railroad  erected  in  Park  Ave.,  New  York  City,  1893.*  It  rests  on 
three  rows  of  columns,  carrying  three  steel  plate  girders  of  about  65  feet  average  span, 
and  of  a  depth  of  7  feet.  The  columns  are  bolted  to  broad  cast-iron  bases,  which  in  turn 
rest  on  masonry  piers,  so  that  they  may  be  considered  as  fixed  at  the  ends.  At  top  they  are 
braced  in  all  ways  by  means  of  curved  plate  brackets. 

There  are  neither  cross-girders  nor  cross-ties  properly  speaking.  The  structure  is  braced 
laterally  by  means  of  a  lattice  construction  between  the  main  girders  on  the  deck  portion  and 
by  the  column  brackets  and  gusset  knee-braces  on  the  through  portion.  The  floor  is  of  solid 
box-construction,  made  up  of  f-inch  steel  plates,  the  rectangular  elements  being  17  inches  high 
and  15  inches  wide.  These  extend  entirely  across  the  whole  floor  and  rest  on  the  three  gir- 
ders in  the  deck  portion  and  are  riveted  between  the  girders  on  the  through  portion.  This 
forms  a  solid  steel  floor,  on  which  the  rails  rest  directly  as  shown  in  the  plate.  The  floor  acts 
as  so  many  eye-beams,  or  plate  cross-girders,  having  a  height  of  17  inches,  and  f-inch  webs 
placed  every  15  inches  throughout  the  entire  length  of  the  structure. 

This  road  will  carry  some  five  hundred  trains  a  day  upon  its  four  tracks,  and  is  supposed  to 
embody  the  latest  and  best  practice  in  elevated  railways  for  standard  traffic.  The  space  between 
the  column  bases  and  the  cast-iron  wheel-guards  is  filled  with  Portland-cement  mortar.  The 
drawings  are  so  complete  that  is  not  necessary  to  describe  them  further. 

414.  Economical  Span  Lengths  and  Approximate  Cost. — The  cost  of  different  kinds 
of  construction  varies  but  little,  if  each  kind  is  designed  for  maximum  economy.  The  eco- 
nomical span  lengths  vary.  For  plate-girder  construction  a  span  length  of  from  40  to  50  feet 
will  result  in  the  greatest  economy  of  total  cost  (including  foundations).  For  riveted  trusses 
the  span  length  should  be  from  50  to  60  feet.  The  economical  and  proper  depths  for  riveted 
trusses  necessitate  raising  the  level  of  the  track,  as  the  clearance  or  height  from  the  street 
surface  to  the  under  side  of  the  iron-work  must  remain  the  same  :  and  this  constitutes  a  very 
important  and  valid  objection  to  this  style  of  construction. 

The  weight  of  iron  in  the  superstructure  of  an  ordinary  well-designed  double-track  ele- 
vated railroad  structure  for  passenger  traffic  varies  from  2000  to  2500  tons  per  mile,  and  the 
cost  per  mile  of  such  a  structure,  including  superstructure,  track,  station  buildings,  and  foun- 
dations, varies  from  $180,000  to  $225,000.  The  cost  per  mile  of  a  standard  double-track 
elevated  railway  suitable  for  heavy  freight  traffic,  including  foundations,  but  not  including 
station  buildings,  will  be  from  $325,000  to  $360,000. 

*  Described  in  Engineering  News,  May  25,  1893.  Walter  Katt6,  M.  Am.  Soc.  C.E.,  Chief  Engineer;  Geo.  H 
Thomson,  M.  Am.  Soc.  C.E.,  Engineer  of  Bridges. 


Plate  XI. 


THE  ESTHETIC  DESIGN  OF  BRIDGES. 


411 


CHAPTER  XXVI. 
THE  ESTHETIC  DESIGN  OF  BRIDGES.* 
INTRODUCTION. 

415.  Development  of  ^Esthetics. — The  character  of  a  people,  a  nation,  or  an  age  is 
manifested  in  its  artistic  productions.  The  forms  which  are  bred  into  the  mind  from  in- 
fancy determine  its  peculiar  taste  for  beauty.  Out  of  the  muhitude  of  nature's  phenomena 
the  human  mind  strives  to  conceive  the  laws  governing  tlieir  relationship.  It  is  possible  to 
formulate  a  clear  and  harmonious  conception  of  the  seemingly  optional  purpose  of  things,  only 
by  comprehending  the  deep  necessities  of  nature's  laws.  Originally  man  had  only  natural 
products  after  which  to  pattern,  and  the  primitive  ideas  of  art  were  improved  through  the 
progress  of  generations  until  the  different  races  of  the  earth  gave  rise  to  the  various  archi- 
tectural forms.  These  forms  are  peculiar  to  the  surroundings,  the  necessities,  and  the  degree 
of  civilization  of  a  people. 

When  we  utilize  the  laws  governing  gravitation  and  the  strength  of  material  for  the  use 
and  convenience  of  man,  we  create  artificially  that  which  nature  has  exemplified.  A  pleasing 
effect  will  be  produced  by  accustomed  forms,  which  shows  that  we  arc  strongly  influenced  by 
surroundings  and  governed  by  our  environment.  When  human  creations  contradict  nature, 
then  they  cease  to  comply  with  our  aesthetic  feeling.  In  the  course  of  architectural  develop- 
ment we  depart  from  the  rudimentary  ideas.  As  long  as  the  mind  keeps  pace  with  this 
departure  it  retains  the  feeling  of  aesthetic  satisfaction.  When  the  simple  form  and  purpose 
of  a  structure  become  so  disguised  that  the  mind  cannot  grasp  them,  then  the  effect  will  be  to 
create  dissatisfaction,  which  is  contrary  to  all  ideas  of  beauty. 

Since  the  civilized  world  lays  a  strong  claim  to  good  taste,  and  since  bridges  are  monu- 
mental to  the  advances  of  civilization,  their  design  may  well  be  subjected  to  the  laws 
governing  aesthetics.  The  theme  of  this  chapter,  therefore,  will  be  to  consider  the  purpose 
of  a  bridge  from  a  higher  standpoint  than  that  of  absolute  necessity,  and  to  characterize  its 
form  with  the  attribute  of  beauty. 

416.  Hindrances  to  Artistic  Design. — There  are  two  considerations  which  singly  or 
combined  would  oppose  artistic  design.  One  of  these  is  given  by  local  conditions,  such  as 
legal  requirements,  inadequate  building  material,  or  unsuitable  location.  These,  when  unavoid- 
able, will  often  excuse  the  engineer  if  his  design  is  not  artistic,  but  in  many  cases  good 
judgment  combined  with  sound  aesthetic  principles  will  aid  in  producing  a  better  result  than 
would  ordinarily  be  obtained.  The  other  and  perhaps  more  common  reason  is  found  in 
financial  considerations.  These  difficulties  will,  however,  not  excuse  the  majority  of  gross 
violations  of  aesthetic  design  which  we  find  everywhere.  It  might  be  in  place  here  to  attempt 
an  explanation  of  some  of  the  foremost  causes  of  such  violations. 

With  the  aim  of  obtaining  cheap  work  competition  is  invited,  resulting  in  favoring  the 
lowest  bidder.  Unless  the  artistic  appearance  of  a  structure  is  imposed  as  a  necessary  feature, 
it  is  rarely,  if  ever,  considered  by  contractors.  Their  sole  object  is  to  satisfy  only  the  absolute 
requirements  of  strength  and  dimensions.    Another  cause  is  the  general  lack  of  good  taste. 

*  Through  an  oversight,  which  is  sincerely  regretted  by  the  authors,  no  credit  was  given  in  the  first  editions  of  this 
work  to  Prof.  R.  Baumeister  for  valuable  suggestions  for  this  chapter  found  in  his  "  Mandbuch  der  Ingenieur-Wissen- 
schaften  ;  "  also  for  the  cuts  in  Plates  XIV,  XV,  and  XVI,  and  for  designs  I  to  21  and  30  to  32  of  Plate  XX. 


412 


MODERN  FRAMED  STRUCTURES. 


It  is  hoped  by  bringing  this  subject  before  American  engineers  to  realize  an  advance  in  the 
aesthetic  design  of  bridges. 

Out  of  the  consideration  that  the  general  public  lays  stress  on  the  appearance  of  its 
bridges  in  cities,  parks,  etc.,  and  often  appropriates  money  to  supply  this  want,  we  realize  an 
advance  in  artistic  taste.  At  the  same  time  the  financial  consideration  is  regarded  as  the 
most  important  feature  in  public  works ;  therefore  the  question  of  aesthetics  is  barred  out  on 
general  principles.  But  rightly  considered,  these  two  factors  are  not  necessarily  contradictory. 
As  above  stated,  there  are  instances  where  the  engineer  is  bound  by  so  many  unavoidable 
restrictions  that  a  pleasing  or  satisfying  result  cannot  be  attained.  But  where  such  limitations 
do  not  exist  and  the  competent  designer  is  left  to  his  own  discretion,  the  primary  form  may  as 
well  be  artistic  as  otherwise.    It  costs  nothing  to  display  good  taste. 

FUNDAMENTAL  PRINCIPLES. 

417.  Artistic  Analysis. — While  the  object  of  structural  analysis  demands  scientific 
perfection  of  every  material  consideration,  the  artistic  analysis  consists  in  supplying  aesthetic 
satisfaction  or  beauty  of  appearance.  The  superiority  of  artistic  over  simple  natural  beauty 
is  due  to  the  clearer  display  of  the  laws  governing  taste.  We  will  classify  the  primary  ideas 
underlying  aesthetic  design  in  the  order  of  their  importance : 

418.  Symmetry  is  the  fundamental  idea  of  aesthetics.  The  symmetrical  order  of  a  series 
of  spans  with  respect  to  a  centre  line  gives  the  impression  of  clearness  of  principle.  The 
mind  finds  no  cause  to  question  the  general  disposition  (see  PI.  XXIII).  Spans  of  different 
lengths  placed  without  regard  to  symmetry  would  call  forth  criticism  (Pi.  XXII).  When  a 
channel  span  does  not  fall  in  the  middle  of  a  river,  or  the  foundations  compel  unfortunate 
location  of  piers,  in  each  case  the  cause  for  unsymmetrical  disposition  remains  hidden.  Where 
the  form  clearly  shows  its  adaptation  to  the  natural  profile  the  laws  of  symmetry  may  be 
violated  without  disturbing  the  good  general  effect  (Fig.  2,  PI.  XXIV).  If,  however,  the 
possibility  for  symmetrical  balance  is  visible,  the  result  will  be  unfavorable  (see  PI.  XXII). 

419.  The  Style  of  a  Structure  should  be  in  conformity  with  the  surrounding  landscape. 
A  bridge  in  a  wild  surrounding  of  rocks  and  forests  over  a  swift  mountain  stream  should  be 
bold  in  appearance.  The  masonry  should  be  coarse,  unfinished,  only  showing  clearly  the 
traces  of  artistic  order  in  contradistinction  to  nature's  wilds  (Fig.  2,  PI.  XXVII).  If  the  rocky 
cliffs  be  very  massive,  the  structure  should  accordingly  convey  that  impression.  In  such  a 
landscape  nothing  can  be  more  appropriate  than  a  heavy  masonry  arch. 

There  should  be  no  doubt  as  to  the  relative  importance  of  landscape  or  bridge.  If  the 
latter  be  large  in  proportion,  it  should  stand  out  in  relief,  robbing  the  landscape  of  its 
supremacy  (Figs.  l  and  2,  PI.  XXI,  and  Fig.  i,  PI.  XXXI).  On  the  other  hand,  if  the  bridge 
be  comparatively  insignificant,  then  it  is  proper  rather  to  underestimate  its  value.  The  result 
will  be  decidedly  in  favor  of  producing  a  pleasing  efYect. 

In  a  thinly-wooded  surrounding,  as  in  a  public  park,  a  structure  should  also  have  a  graceful 
and  more  finished  appearance  (Fig.  i,  PI.  XXVII).  If  it  is  a  city  bridge  in  the  vicinity  of 
immense  buildings,  perhaps  of  beautiful  designs,  then  we  must  look  to  a  majestic  harmony 
with  the  prevalent  style  of  the  neighboring  architecture  (see  Pis.  XXV  to  XXXI,  and  details 
on  Pis.  XIV  to  XVII).  The  masonry  will  consist  mostly  of  cut  stone  to  give  the  neat  and 
clean  outline  characteristic  of  the  city. 

In  a  general  way,  we  must  be  careful  not  to  carry  any  one  feature  to  the  extreme,  as  this 
would  destroy  the  harmony  and  thus  detract  from  the  general  appearance. 

420.  The  General  Form  should  never  disguise  the  purpose  of  a  structure,  but  should 
aid  in  impressing  the  mind  with  visible  strength  and  proper  adaptation  to  purpose. 


THE  ESTHETIC  DESIGN  OF  BRIDGES. 


413 


Similar  means  must  be  employed  to  accomplish  similar  ends.  This  maxim  establishes 
order  and  purpose  of  form.  A  bridge  consisting  of  several  masonry  arches  and  several  metal 
spans  of  equal  lengths  will  not  appear  harmonious  in  design  unless  these  are  properly  grouped 
(see  Figs.  7  and  9,  PI.  XXII).  Even  then  the  question  arises,  why  would  not  the  same  material 
have  answered  for  all  the  spans?  Certainly  in  such  a  case  it  would  be  more  in  accordance  with 
good  taste  to  place  longer  metal  spans  in  the  centre  or  most  important  part  of  the  bridge,  and 
to  mark  distinctly  the  lesser  importance  of  the  adjoining  masonry  arches,  as  on  PI.  XXIII. 
This  divides  up  the  structure  into  main  bridge  and  approaches. 

The  artistic  form  strives  to  make  clear  the  relation  between  the  different  members;  not 
only  to  emphasize  certain  ones,  but  to  show  the  dependence  of  one  upon  another.  This 
explains  the  preference  of  artistic  form  to  sober  design.  When  the  static  principles  under- 
lying a  structure  are  inconceivable  to  the  public  mind,  the  latter  will  not  be  impressed  with 
the  idea  of  safety  or  beauty.  In  such  instances  the  safety  is  actually  demonstrated  by  tests 
which  are  often  intended  to  win  over  public  prejudice.  It  is  therefore  easy  to  understand 
why  the  recent  developments  in  engineering  are  not  generally  considered  beautiful.  No 
matter  what  technical  advantages  they  may  possess,  we  must  accustom  ourselves  to  the  new 
form  and  develop  a  liking. 

A  stream  spanned  by  a  bridge  of  less  length  than  the  breadth  of  stream  would  make  the 
structure  appear  inadequate  (Fig.  5,  PI.  XIV;  Figs,  i  and  4,  PI.  XVII).  The  main  spans  may 
be  carried  past  the  banks  of  a  river,  thus  giving  an  unnecessary  importance  to  the  bridge.  We 
readily  fall  upon  these  difficulties  of  design  where  the  water-level  is  subject  to  great  changes. 
It  is  plain  then  that  any  omission  or  any  unnecessary  addition  would  tend  to  belie  the  form 
of  its  purpose. 

Let  it  be  generally  understood  that  in  no  case  should  the  artistic  ideas  governing  form 
contradict  any  static  consideration.  Both  have  their  origin  in  the  same  natural  laws,  and 
when  they  conflict  the  static  correctness  should  always  be  preferred  to  beauty. 

421.  The  Dimensions  of  the  various  parts  of  a  structure  should  bear  a  harmonious  rela- 
tion to  the  whole.    This  constitutes  harmony  of  proportions. 

For  a  given  length  of  a  bridge,  all  things  considered,  there  will  be  certain  lengths  for  the 
various  spans  which  will  give  the  best  general  effect.  Again,  the  subdivision  of  a  span  into 
panels  should  bear  a  certain  relation  to  both  its  height  and  length.  So  also  a  railing  should 
consist  of  panels  equal  to  a  definite  fraction  of  the  truss  panel. 

All  these  questions  are  usually  determined  from  the  statical  considerations  of  the  problem, 
but  frequent  opportunity  is  given  to  vary  such  proportions  in  favor  of  good  appearance  with- 
out interfering  materially  with  the  technical  or  practical  conditions.  We  are  often  compelled 
to  dimension  a  secondary  member  unnecessarily  large  just  for  the  sake  of  looks,  which  can 
easily  be  done  by  choosing  a  flat  or  angle  instead  of  a  rod. 

When  cast-iron  or  wood  is  used  for  a  column  we  commonly  find  abnormal  forms.  The 
rest  of  the  structure  being  wrought  will  require  much  less  material  and  the  result  will  be  a  lack 
of  harmony  in  dimensions. 

422.  Ornamentation  should  be  regarded  as  distinct  from  simple  or  rudimentary  form 
and  should  be  utilized  only  to  reinforce  the  same. 

It  is  not  the  object  of  ornamentation  to  change  the  general  character  of  a  structure  to 
such  an  extent  as  to  hide  the  underlying  principles.  Its  proper  application  is  to  aid  the  mind 
in  the  conception  of  the  purpose,  by  contrasting  different  members  of  a  structure  with  each 
other.  This  law  is  so  frequently  violated  that  we  are  often  at  a  loss  whether  or  not  to  believe 
what  we  see.  Covering  a  framework  with  boards  and  paint  to  imitate  a  stone  arch  ;  a  hori- 
zontal girder  blended  with  arch  construction  ;  a  foot-bridge  modelled  after  a  Roman  aqueduct; 
or  decorating  a  girder  to  represent  a  temple,  which  latter  hangs  in  the  air  instead  of  resting 


414 


MODERN  FRAMED  STRUCTURES. 


on  the  ground — all  these  are  adulterations  and  belie  the  underlying  principles  by  exhibiting 
false  forms. 

When  we  ornament  the  ring  of  an  arch,  emphasize  the  line  of  a  roadway,  or  surmount  a 
pier  by  a  statue  or  other  decoration  to  mark  the  importance  of  such  parts,  we  add  to  the 
clearness  and  hence  to  the  aesthetic  appearance. 

This  subject  will  be  taken  up  in  detail  in  the  articles  on  Ornamentation. 

INFLUENCE   OF  BUILDING   MATERIAL,   COLOR,   AND   SHADES  AND  SHADOWS  ON  THE 

.ESTHETIC  APPEARANCE  OF  A  BRIDGE. 

423.  Material. — One  of  the  foremost  considerations  upon  which  the  popularity  of  a 
structure  depends  is  the  building  material.  Stone  occupies  the  highest  rank,  as  it  possesses 
properties  requisite  to  aesthetics  which  are  peculiar  to  no  other  material.  As  a  consequence 
of  its  low  unit  strength  and  the  necessary  manner  of  bonding,  we  obtain  massive  forms  com- 
pared to  wood  or  metal.  To  the  average  mind,  stone  lends  the  appearance  of  solidity  and 
strength,  and  emphasizes  the  monumental  character  of  architecture.  The  general  public  is 
most  familiar  with  the  relation  existing  between  the  weight  and  strength  of  this  material,  so 
that  even  the  uneducated  eye  can  discern  whether  a  structure  seems  unusually  bold  or  heavy. 
Wood  is  not  sufficiently  durable  for  permanent  structures  and  is  not  considered  here. 

Iron  and  steel  are  in  strong  contrast  to  stone.  The  relation  of  strength  to  weight  in  iron 
gives  rise  to  such  light  forms  that  the  mind  is  often  impressed  with  wonder.  Metal  designs 
display  such  light  masses  compared  to  masonry  that  it  is  difficult  to  accustom  ourselves  to 
this  material.  Another  fact  which  speaks  in  favor  of  stone  is  that  it  occurs  nearly  every- 
where in  nature,  whereas  metallic  iron  is  a  manufactured  product.  These  differences  in  the 
choice  of  material  are  inherent  in  the  customs  and  surroundings  of  a  people. 

When  the  application  of  iron  and  steel  to  bridge-building  has  become  characteristic  of 
the  architecture  of  a  nation,  then  this  material  will  be  popular  from  an  aesthetic  standpoint. 

Let  it  not  be  supposed  from  the  preceding  that  the  choice  of  artistic  form  is  limited 
by  the  material.  It  is  possible  to  symbolize  the  same  natural  laws  in  almost  any  material. 
A  column,  whether  of  stone,  wood,  or  iron,  will  not  change  its  character.  When,  however, 
artistic  form  is  applied  to  construction,  the  material  is  intimately  related  thereto.  For 
example,  the  capital  and  base  of  a  stone  column  should  receive  different  dimensions  and  shapes 
from  those  of  a  cast-iron  column. 

We  cannot  realize  absolute  stability,  or  an  ideal  balance  between  loads  and  forces,  for 
these  are  relative  properties  varying  with  the  kind  of  material.  In  viewing  a  structure  we 
involuntarily  infer  its  material  from  its  form  even  if  paint  has  been  utilized  to  disguise  natural 
color.  Artistically  considered,  it  is  improper  to  imitate  a  stone  ornament  in  tin  or  wood, 
as  the  strength  and  weight  of  the  latter,  as  also  the  manner  of  workmanship,  contradict 
such  forms.  Even  if  the  progress  of  the  arts  should  make  it  possible  to  produce  any  or  all 
forms  out  of  any  material,  we  would  regard  such  progress  wrongly  applied  when  the  attempt 
is  made  to  destroy  characteristics  rather  than  to  emphasize  them.  Such  disregard  of  the 
natural  properties  of  material  would  eventually  lead  to  coarse,  miniature  forms,  lacework  on 
an  abnormal  scale,  or  even  to  an  esthetic  lie.  Technical  miracles  have  no  claim  to  beauty, 
since  they  lack  harmony  of  form. 

The  general  effect  is  most  likely  to  be  pleasing  when  but  one  kind  of  material  is  used 
(see  Figs,  i,  2,  3,  and  5,  PI.  XXI ;  Figs,  i,  2,  and  3,  PI.  XXIV  ;  and  Fig.  i,  PI.  XXXI).  Grf  at 
diversity  in  the  static  balance  or  in  the  technical  execution  of  members,  especially  when  thc-se 
are  of  different  materials,  compel  the  spectator  to  change  the  scale  with  which  he  measures  the 
relative  magnitude  of  forces  (see  Figs.  2  and  9,  PI.  XXII).  He  may  find  it  difficult  or  even 
impossible  to  obtain  a  general  harmonious  impression.    Such  examples  are  numerous;  as  when 


THE  ESTHETIC  DESIGN  OF  BRIDGES. 


415 


masonry  arches  and  iron  superstructure  of  same  length  of  span  are  combined  in  one  bridge; 
or  a  brick  arch  between  metal  spans  ;  or  ornaments  made  of  different  material  from  the  main 
structure.  On  the  other  hand,  it  is  often  possible  to  obtain  rather  a  fortunate  result  by 
varying  the  material,  as  in  a  light  metal  span  between  masonry  approaches  (see  PI.  XXIII), 

Both  the  laws  of  harmony  and  of  contrast  are  entitled  to  their  places  in  art.  It  is  not, 
however,  admissible  to  hide  or  ignore  a  member  with  the  view  of  producing  uniformity  of 
effect.  For  example,  a  rod  used  to  take  up  the  horizontal  thrust  of  an  arch,  to  cover  the  lack 
of  stability  of  supporting  columns.  Instead  of  giving  this  member  its  architectural  significance) 
thin,  thread-like  rods  are  frequently  employed.  When  we  fail  to  comply  with  such  aesthetic 
wants,  we  allow  an  unfortunate  sense  of  wonder  to  fill  the  lack  of  technical  truthfulness.  We 
dispense  with  the  solution  of  one  of  the  most  important  problems  in  architecture. 

424.  Color. — It  is  a  well-known  fact  that  colors  produce  an  impression  on  the  mind 
which  may  be  pleasing  or  otherwise,  quite  similar  to  the  effect  of  a  musical  chord  or  discord 
upon  the  ear.  An  object  may  appear  in  perfect  harmony  with  its  surroundings,  but  when 
placed  elsewhere  would  lose  its  entire  beauty.  It  would  seem  then  that  the  question  of  color 
should  be  regarded  as  one  of  considerable  weight  in  the  artistic  design  of  an  engineering 
structure.  Hitherto  this  matter  has  been  left  to  the  developed  taste.  Since  our  vision  does 
not  afford  us  the  same  aid  in  choosing  color  that  we  obtain  from  statics  and  dynamics  in  the 
choice  of  form,  it  would  seem  proper  to  add  this  feature  to  practical  a;sthetics. 

All  colors  may  be  divided  into  two  classes,  primitive  and  collective.  The  former  class 
comprises  the  rainbow  colors,  while  the  latter  are  primary  colors  diluted  with  white  or  gray. 
As  may  be  supposed,  this  difference  is  not  physically  marked,  but  is  of  purely  aesthetic  character. 
In  the  scale  we  ascend  from  the  primitive  colors,  blue,  yellow,  and  red,  to  their  primitive  com- 
binations, violet,  green,  and  orange,  and  finally  to  the  collective  color  white  with  its  gray 
variations.  Both  classes  have  admirers.  The  masses  are  more  inclined  to  favor  the  intense 
pigments,  while  the  educated  minority  show  a  preference  for  the  softer  shades.  This  is  not 
the  result  of  a  lack  of  appreciation  for  the  bright  colors,  but  of  the  development  of  taste. 

The  choice  of  color  should  be  in  accordance  xvitJi  the  material  to  be  covered.  We  avoid  the 
use  of  paint  when  it  is  not  necessary  to  the  preservation  of  the  material.  Stone  occurs  in  such 
variety  of  tints  and  requires  no  preservative,  so  that  we  can  usually  satisfy  our  wants  without 
resorting  to  artificial  means.  Iron  and  wooden  parts,  therefore,  would  involve  the  question 
of  paint.  When  a  single  color  is  used,  let  this  be  a  soft  tint  in  harmony  with  the  stone  and 
other  surroundings.  Different  shades  may  be  employed  only  to  reinforce  the  static  principles 
of  a  structure.  It  would,  however,  be  decidedly  against  good  taste  to  imitate  material  by  the 
use  of  paint.  An  iron  structure  disguised  to  represent  wood  cannot  produce  a  pleasing 
effect,  since  the  form  would  appear  too  light.  The  same  with  the  reverse,  when  the  form 
would  seem  abnormally  heavy. 

In  suiting  the  color  to  the  surrotmding  landscape  the  question  of  relative  magnitude  or 
importance  is  greatly  involved.  If  the  bridge  is  small,  it  may  be  well  to  effect  a  contrast  or 
relief  by  the  choice  of  complementary  colors.  As  in  a  wooded  locality,  red  or  white  would 
be  well  chosen.  If,  on  the  other  hand,  the  structure  is  so  large  that  the  landscape  appears 
comparatively  insignificant,  a  wide  range  is  offered  without  incurring  the  loss  of  contrast. 

Coloration  should  emphasise  the  static  differe7ices  of  the  main  parts.  In  some  instances  it 
may  be  desirable  to  employ  various  colors  to  aid  in  the  decoration  of  a  structure.  Then  the 
paint  becomes  part  of  the  ornamentation.  Heavy  masses  or  the  principal  members  should  in 
this  case  receive  a  heavy  color,  while  members  carrying  little  or  no  load  should  be  made  to 
appear  slender.  Also  a  lack  of  relative  proportion  may  be  partly  overcome  by  this  means. 
We  designate  a  color  as  being  heavy  when  it  seemingly  increases  the  size  of  ar.  object. 
Livelier  shades  may  be  employed  for  members  which  characterize  the  features  ot  outlines  6, 
a  bridge.    More  subdued  tints  would  be  applicable  to  the  less  conspicuous  s,.rfaccs.  Mem» 


4x6 


MODERN  FRAMED  STRUCTURES. 


bers  used  in  the  same  capacity  should,  however,  be  treated  alike.  There  is  only  one  admissible 
exception  to  this  rule,  as  when  we  vary  two  colors  in  regular  succession.  The  stones  of  an 
arch  ring  may  alternate  red  and  gray  without  destroying  the  clearness  of  purpose.  The 
beauty  of  a  structure  is  increased  by  ornamentation,  but  let  it  be  remembered  that  this  beauty 
may  be  easily  destroyed  by  extremes  and  by  overloading. 

The  general  color  effect  should  agree  with  the  character  of  a  structure.  Bridges,  as  a  rule, 
demand  a  more  sedate  appearance,  for  which  reason  it  is  an  easy  matter  to  exceed  the  bounds 
of  good  taste.  Grace  and  neatness  may  be  displayed  by  resorting  to  different  shades  for 
railings,  smaller  ornaments,  or  even  for  slender  rods,  as  the  suspenders  or  hangers  of  a  suspen- 
sion bridge.  But,  after  all,  we  are  compelled  to  restrict  the  diversity  of  color  to  contrasting 
only  the  main  parts,  as  superstructure  to  masonry,  etc.  The  tendency  should  be  always  to 
complete  the  general  effect  by  supplying  the  missing  features,  thus  bringing  about  harmony 
between  color  and  the  character  of  a  structure. 

We  attribute  certain  properties  to  colors,  as  warm,  cold,  light,  heavy,  bright  or  lively,  and 
subdued,  all  of  which  are  derived  from  impressions  made  upon  the  mind.  These  terms  are 
explanatory  in  themselves,  but  it  is  well  to  remember  that  red  and  yellow  always  produce 
warmth,  while  blue,  purple,  and  neutral  tint  do  the  opposite.  A  heavy  color  seemingly 
increases  the  size  of  an  object  by  contrast,  a  light  one  tends  to  diminish  apparent  volume  by 
destroying  contrast.  Bright  and  lively  are  applied  to  the  primitives,  and  collective  colors  are 
usually  subdued  shades  of  the  former.  We  utilize  these  properties  in  bringing  about  aesthetic 
unison  between  a  structure  and  its  surroundings,  since  these  should  possess  the  same  character. 

425.  Shades  and  Shadows. — What  has  been  said  of  color  also  applies  to  shades  and 
shadows  to  a  very  great  extent.  All  shapes  and  forms  would  be  destitute  of  any  modulation 
and  would  appear  as  planes  were  it  not  due  to  the  shading  produced  by  one  part  projecting 
over  another.  Upon  this  fact  depend  all  principles  of  ornamentation.  In  designing  copings, 
etc.,  we  must  consider  the  shadows,  as  the  whole  form  is  characterized  by  them.  Smooth-face 
stones  when  laid  would  appear  as  one  mass  were  the  joints  not  of  another  color,  whereas  in 
rough-face  stone  the  joints  are  indicated  by  shadows.  Great  art  may  be  displayed  in  this 
feature  of  a  design,  but  the  good  result  is  too  often  left  to  chance. 

ORNAMENTATION. 

426.  In  Bridge  Building  little  opportunity  is  afforded  to  give  members  their  ideal  archi- 
tectural significance,  because  their  forms  are  determined  by  the  technical  requirements  ol 
strength,  durability,  and  cost.  We  may  well  be  satisfied  when  the  general  outline  of  a  bridge 
exhibits  a  graceful  and  pleasing  appearance,  as  in  Pis.  XXI  and  XXII.  Only  in  particular 
cases,  as  for  arch  rings,  portals,  piers,  cornices,  and  railings,  we  are  enabled  to  develop  artistic 
forms.  (See  Pis.  XIV  to  XVII.)  Therefore,  to  complete  the  aesthetic  value  of  a  structure,  it  is 
proper  to  elaborate  these  decorative  elements  by  supplying  ornaments.  In  so  doing  we  must 
consider  the  quantity,  distribution,  scale,  choice,  and  style  both  of  the  ornaments  and  of  the 
structure  itself. 

427.  Quantity  of  Decoration. — It  is  contrary  to  the  principle  of  aesthetic  economy  to 
overload  a  structure  or  any  part  thereof  with  ornaments,  even  if  these  details  are  pleasing  and 
properly  chosen.  The  result  would  be  to  suppress  or  disguise  the  purpose  of  the  main 
members  and  to  exhibit  an  unbecoming  wastefulness.  The  plain  or  elaborate  character  of  an 
entire  structure  must  not  be  contradicted  by  any  of  its  parts.  In  this  branch  of  engineering 
there  is  little  danger  of  extravagance,  but  rather  a  lamentable  insufficiency.  The  relation 
between  a  structure  and  its  decorations  should  impress  the  observer  with  aesthetic  satisfaction. 

428.  The  Distribution  of  Ornaments  maybe  utilized  to  emphasize  certain  parts  without 
opposing  the  artistic  perfectness  of  a  bridge.  The  more  important  river  spans  would  deserve 
being  ornamented  rather  than  their  approaches ;  likewise  an  arch  rather  than  its  wing-  and 


THE  ESTHETIC  DESIGN  OF  BRIDGES. 


417 


retaining-walls.  (See  Pis.  XIV  to  XVII.)  Buttresses  may  be  more  elaborate  than  the  wall 
to  which  they  belong. 

As  a  rule,  members  carrying  heavy  loads  should  receive  less  decoration  than  those  taking 
little  stress.  Hence  the  base  of  a  pier  would  be  plain  compared  to  its  coping  or  capital  (Figs. 
3,  4,  6,  and  9,  PI.  XVI) ;  so  also  main  trusses  relatively  to  the  railings  and  cornices  (Figs.  6  and 
7,  PI.  XIV  and  PI.  XXIII).  The  parts  situated  closer  to  the  observer  are  more  valuable 
aesthetically  than  distant  ones,  for  which  reason  the  end  portals,  railings,  etc.,  should  be 
ornamented.    (Examples  in  Pis.  XIV  to  XVII.) 

On  the  other  hand,  such  contrasts  must  not  be  too  strong.  A  statue  would  seem  out  of 
place  on  a  rough  block,  or  fine  ornamental  details  on  an  otherwise  plain  plate  girder.  The 
attempt  to  attain  a  proper  equilibrium  between  the  architecture  of  a  portal  and  the  adjoining 
bare  superstructure  has  often  been  unsuccessful.  The  tendency  is  to  give  a  preference  to  the 
former,  especially  when  cheap  cast  forms  are  at  hand,  while  the  comparative  treatment  of  the 
latter  remains  neglected.  Many  designers  have  given  way  to  the  temptation  of  elaborating 
in  cast-iron,  while  leaving  wrought  metal  in  its  plain  and  unfinished  condition.  It  shows  poor 
taste  in  design  when  we  find  neat  columns  or  piers  supporting  homely  roof  trusses,  plate 
girders,  etc. 

429.  Scale  of  Ornamentation. — The  dimensions  of  a  bridge  determine  the  relative  sizes 
of  its  ornaments.  Small  decorations,  even  in  large  quantities,  are  not  suitable  to  huge 
structures,  and  vice  versa.  It  is  difficult  to  attain  a  happy  medium  in  this  important  feature, 
because  a  bridge  is  subject  to  examination  from  two  main  points  of  observation :  the  one 
away  from  the  structure,  from  which  the  general  appearance  would  be  judged  ;  the  other  on 
the  roadway,  from  which  portals,  trusses,  railings,  etc.,  would  be  viewed.  We  depend  upon 
artistic  tact  for  the  preservation  of  clearness. 

When  comparison  is  possible  between  two  objects  of  similar  form,  though  differing 
widely  in  dimensions,  the  seeming  adaptation  to  purpose  will  be  disturbed.  Small  forms 
should  never  be  a  mere  reduction  of  large  ones  when  both  are  applied  to  the  same  structure. 
Even  when  the  same  purpose  is  accomplished  by  each,  it  is  well  to  modify  the  former,  usually 
by  omitting  some  of  the  details  of  the  latter.  We  see  such  examples  where  the  capital  or 
head  of  a  pier  is  repeated  in  the  small  columns  of  the  railing,  or  when  spans  of  same  design, 
differing  greatly  in  length,  succeed  each  other  in  a  bridge  (Figs.  7  and  10,  PI.  XXII).  In  a 
few  instances  the  repetition  of  a  form  in  various  sizes  is  admissible  when  the  material  is 
changed  ;  as  a  long  metal  arch  with  adjoining  masonry  arches.    (See  PI.  XXIII.) 

We  must  not  forget  that  all  structures,  even  the  most  gigantic,  are  erected  by  and  for 
man ;  hence  statuary,  columns,  and  all  other  ornaments  shaped  to  represent  objects  belonging 
to  the  animal  or  vegetable  kingdom  should  approach  the  natural  dimensions.  The  impression 
made  by  exaggerated  artistic  forms  is  not  one  of  intellectual  greatness,  but  simply  a  material 
hugeness,  wherein  grace  and  delicate  details  have  been  lost. 

430.  Choice  of  Artistic  Form. — The  perfect  artistic  form  of  an  object  should  recall  the 
structural  purpose  of  its  parts  without  necessitating  close  observation  or  analysis.  The 
strength  of  a  member,  its  direction,  application,  and  manner  of  resisting  forces,  should  be 
readily  comprehensible.  It  is  easiest  to  illustrate  this  idea  by  citing  some  of  the  many  gross 
violations  where  attempts  are  made  to  decorate  but  without  success.  On  some  of  the  old 
covered  bridges  the  wooden  suspenders  have  been  disguised  to  represent  compression  columns, 
by  supplying  bases  and  capitals.  On  the  Callowhill  Street  Bridge  in  Philadelphia,  the 
main  trusses  take  the  shape  of  an  arched  passage,  the  vertical  struts  acting  as  piers,  with  the 
top  chord  representing  a  series  of  arches.  Adding  a  frame-like  decoration  to  wall  spaces 
between  buttresses  is  out  of  place,  because  the  forces  to  be  resisted  have  no  bearing  on  such 
forms.  The  plain  surface  of  a  beam  or  girder  is  not  appropriate  for  a  frieze-like  ornament 
extending  from  end  to  end.    The  centre  of  the  beam  should  be  marked  by  some  special  feat* 


MODERN  FRAMED  STRUCTURES. 


ure,  from  which  the  decoration  may  extend  towards  the  ends.  It  is  also  advisable  to  choose 
ornamental  details  involving  no  particular  direction,  as  isolated  figures  or  panels  (Fig.  6,  PI. 
XIV,  also  Pis.  XVIII  and  XIX). 

431.  Style. — The  chief  end  of  art  is  not  attained  by  copying  exactly  the  forms  of  nature 
or  those  of  textile  fabrics,  but  by  reproducing  the  analogy  between  the  static  functions  of 
natural  products  to  satisfy  artificial  wants.  Exact  imitations  are  limited  in  architecture  by 
difificulties  relating  to  their  execution  in  stone,  wood,  or  metal.  The  objects  to  be  accom- 
plished by  style  are;  to  prepare  the  natural  forms  for  artistic  purposes  ;  to  maintain  the  struct- 
ural elements,  but  supply  the  necessary  coarseness  to  correspond  with  the  building  material ; 
and,  lastly,  to  neglect  extremely  fine  details  by  changing  the  scale  which  might  slightly 
exceed  the  natural.  In  other  words,  we  create  style  by  a  liberal  translation  of  nature's 
forms. 

The  extent  to  which  style  may  be  applied  depends  upon  the  magnitude  and  location  of 
the  member,  as  on  the  material  and  character  of  the  whole  structure.  Every  age  adopts  a 
style  according  to  its  peculiar  conception  of  nature,  which  explains  the  great  divergence  in 
architectural  forms  while  the  constructive  principles  have  remained  the  same. 

All  that  has  been  said  on  the  subject  of  ornamentation  is  really  defining  "artistic 
beauty ;"  and  after  all  there  is  nothing  that  will  be  of  more  service  to  the  designer  than  a 
well-developed  taste  for  that  which  is  beautiful.  Artistic  taste  may  be  acquired  by  close 
observation  and  study ;  but  even  then  not  everybody  is  susceptible  to  a  broad  understanding 
of  "aesthetics."  We  are  easily  influenced  by  new  impressions,  and  especially  by  the  arts  from 
other  countries. 

Within  the  limits  of  our  environment  our  ideas  of  beauty  naturally  assume  a  domestic 
character.  Every  other  country  has  its  own  style,  and  when  we  become  familiar  with  each 
we  begin  to  realize  that  our  domestic  ideas  are  only  primitive.  Each  style  has  claims  to 
beauty  peculiar  to  itself,  and  we  must  learn  to  value  them,  since  they  are  as  justifiable  as  the 
claims  of  our  own  style. 

ESTHETIC  DESIGN. 

432.  Division  of  Subject. — What  has  been  said  in  the  previous  articles  is  more  of  a  gen- 
eral character.  With  the  aim  of  increasing  the  value  of  the  present  chapter  we  will  consider 
briefly  the  design  of  the  principal  elements  of  a  bridge  with  respect  to  general  aesthetic  form 
and  ornamentation.  In  doing  so  the  most  natural  subdivision  of  the  subject  would  be  into 
Substructure y  Superstructure,  and  Roadway. 

433.  Substructure  :  Piers  and  Abutments. — We  divide  up  a  site  into  a  number  of 
spans,  the  points  of  division  being  marked  by  piers,  and  the  extreme  points  by  shore  piers  or 
abutments.  The  piers  serve  the  purpose  of  supporting  the  superstructure,  of  resisting  the 
horizontal  thrust  resulting  from  wind  and  moving  loads,  and  (when  located  in  water)  of  with- 
standing wave  and  current  action.  From  static  considerations  these  forces  will  be  resisted  by 
a  pier  of  certain  form  ;  but  when  the  question  of  beauty  is  involved,  we  are  usually  compelled 
to  modify  this  form  beyond  the  requirements  of  absolute  necessity. 

The  relation  between  height  and  thickness  should  be  in  aesthetic  harmony  with  the  length 
of  span  and  apparent  load  to  be  carried.  For  long  spans  and  small  clearance  the  supports 
should  convey  the  idea  of  massiveness  (Fig.  5,  PI.  XXI).  In  the  other  extreme  of  short  spans 
and  high  piers  a  slender  design,  as  an  iron  bent  or  tower,  would  seem  proper  (Fig.  4,  PI.  XXI). 
A  very  high  viaduct  may  even  be  constructed  with  single  bents  resting  on  pins,  top  and  bot- 
tom, the  horizontal  thrust  being  taken  up  by  fixed  towers  placed  at  regular  intervals.  Such 
a  disposition  would  add  grace  and  \ariety  to  the  structure. 

434.  The  Scale  for  Details  must  be  deduced  from  the  general  dimensions.  A  pier  is 
composed  of  three  parts — base,  body,  and  capital  or  coping.    .^Esthetic  stability  is  added  by 


THE  ESTHETIC  DESIGN  OF  BRIDGES. 


419 


widening  out  the  base.  In  rivers  where  the  stage  of  water  is  variable  it  is  advisable  to  start 
the  base  a  little  above  high  water  and  spread  with  the  increasing  depth,  thus  retaining  a 
visible  base  for  all  conditions  (Fig.  5,  PI.  XXI).  The  base,  being  most  subject  to  the  action  of 
the  external  forces,  should  not  be  decorated.  The  only  admissible  addition  might  consist  of 
a  coping  to  separate  the  base  from  the  body.  This  is  desirable,  for  it  idealizes  the  architect- 
ural significance  of  the  supporting  power  of  a  foundation  (Fig.  3a,  PI.  XIV). 

435.  The  Body  of  a  Pier  is  subject  to  many  variations.  The  general  style  should  agree 
with  that  of  its  base  and  coping.  A  batter  is  very  desirable,  as  it  emphasizes  the  stability 
against  horizontal  forces.  Offsets  in  the  vertical  Hnes,  to  replace  batter,  are  to  be  avoided, 
since  expense  is  'the  only  motive  to  prompt  such  design.  Ornamentation  should  tend  in  a 
vertical  direction,  but  it  is  well  to  maintain  clearness  of  form  and  purpose  by  choosing  artistic 
shapes,  using  decorations  only  to  a  very  limited  extent  (Figs.  3,  4,  and  5,  PI.  XIV  and  PI. 
XVII).  When  the  body  is  carried  above  the  supports,  as  for  an  arch  or  deck  bridge,  this  point 
should  be  characterized  by  some  form  of  cornice  or  coping.  The  portion  above  the  supports 
is  of  minor  importance;  it  should  therefore  receive  smaller  dimensions  and  may  be  more 
elaborately  decorated  (Figs,  i  and  2,  PI.  XV  ;  Figs.  3,  4,  and  6,  PI.  XVI ;  Fig.  5,  PI.  XXI). 

436.  The  Coping. — In  treating  the  coping,  which  really  takes  the  place  of  the  capital  of 
a  column,  a  wide  range  both  of  form  and  purpose  is  offered.  The  common  and  most  fre- 
quent style  of  coping  consists  of  a  heavy  course  of  cut  stone  projecting  over  the  body  of  the 
pier.  Besides  distributing  the  load  over  the  area,  it  emphasizes  the  point  where  the  weight  is 
applied  and  adds  a  finish  or  covering. 

The  coping  is  subject  to  more  ornamentation  than  any  other  portion  of  a  pier,  and  may, 
according  to  circumstances,  develop  into  the  form  of  a  complete  capital.  In  some  cases  the 
body  is  carried  to  the  roadway,  where  it  forms  a  prominent  point  of  the  railing.  The  cornice 
and  balustrade  may  then  unite  in  offsetting  the  coping  (Figs,  i  and  6,  PI.  XIV  ;  Figs,  i  and  2, 
PI.  XV;  Figs.  3,  4,  and  6,  PI.  XVI ;  and  Fig.  i,  PI.  XVII). 

For  an  even  number  of  spans  a  pier  will  fall  in  the  bridge  centre.  This  pier  should  be 
more  conspicuous  than  the  others,  both  as  regards  size  and  ornamentation.  The  object  is  to 
mark  the  centre  of  the  structure,  an  important  factor  to  symmetry  and  artistic  effect  (Fig.  5, 
PI.  XXI,  right-hand  pier  is  centre  of  bridge). 

437.  The  Abutment  performs  the  function  of  a  pier,  but  usually  embodies  other  features 
which  necessitate  a  more  massive  form.  Generally  an  abutment  constitutes  the  connecting 
link  between  main  structure  and  approach  or  embankment,  in  which  case  the  massiveness  is 
required  to  serve  the  purpose  of  a  retaining  wall.  Batter  of  the  front  face  adds  greatly  to 
visible  stability.  For  any  wall  resisting  geostatic  pressure  the  theoretical  line  of  the  face  is  a 
curve,  which  answers  the  aesthetic  requirements  best,  as  it  conforms  to  the  popular  con- 
ception of  the  way  in  wliicii  walls  usually  fail,  viz.,  by  bulging  out  or  overturning.  The 
simpler  form  approaching  this  is  obtained  by  a  straight  batter,  while  the  one  with  a  vertical 
front  displays  the  least  stabili'iy,  and  hence  is  wanting  in  artistic  effect. 

Abutments  may  also  be  used  as  office  buildings,  and  then  admit  of  very  elaborate 
designs.  In  case  of  a  suspension  bridge  the  foot  of  a  tower  is  usually  made  to  serve  this  pur- 
pose. When  an  abutment  covers  all  these  requirements  it  may  be  developed  into  the  form 
of  an  end  portal  or  archway.  The  design  of  this  portion  of  a  bridge  is  most  susceptible  to 
architectural  beauty  (Figs.  4  and  7,  PI.  XV,  and  Figs.  1,  2,  5,  and  9,  PI.  XVI). 

It  is  to  be  remembered  that  the  artistic  details  of  piers  and  abutments  should  be  devel 
oped  upward  and  never  down  ;  that  the  coping  takes  precedence  over  the  body,  and  this  over 
the  base,  in  the  order  of  aesthetic  value.  It  is  barely  possible  to  formulate  any  rules  for  the 
details  regarding  the  horizontal  subdivision  of  a  pier  or  other  body  of  masonry  into  courses, 
etc.    These  subdivisions  depend  upon  the  position  of  the  observer  and  the  height,  thickness, 


420 


MODERN  FRAMED  STRUCTURES. 


surroundings,  loading,  material,  shades  and  shadows,  etc.,  of  the  object.  (For  examples  see 
Plates  XIV  to  XVII.) 

438.  Superstructure. — The  question  of  artistic  design  is  more  neglected  in  the  general 
forms  of  trusses  than  in  any  other  portion  of  a  bridge.  The  reasons  previously  mentioned 
for  neglecting  aesthetics  in  bridge  building  bear  more  strongly  on  iron  and  steel  work  than  on 
masonry.  Competition  is  far  greater  among  manufacturers  of  the  former  class,  and  especially 
when  plans  are  prepared  by  them.  The  question  of  extreme  economy  of  both  labor  and 
material  forms  the  basis  for  such  designs.  Therefore,  when  stress  is  laid  on  artistic  effect, 
plans  should  be  prepared  by  the  engineer,  or  else  let  it  be  understood  in  advance  that  design 
as  well  as  cheapness  will  be  considered.  This  will  tend  toward  elevating  the  standard  of 
designing  without  neglecting  economy. 

With  the  possible  exception  of  city  bridges,  ornaments  may  be  entirely  omitted  without 
materially  injuring  the  aesthetic  value  of  a  structure.  The  essential  considerations  are  form, 
symmetry,  and  adaptation  to  surroundings  and  purpose.  When  these  are  complied  with,  so 
that  a  structure  as  a  whole  will  convey  a  pleasing  and  harmonious  effect,  then  we  have  accom- 
plished the  end  of  an  artistic  design. 

To  insure  a  successful  result  aesthetically  considered,  certain  relations  between  the  various 
parts  of  a  bridge  must  be  maintained.  These  have  received  mention  in  the  foregoing,  viz., 
the  proportion  of  height  of  substructure  to  superstructure  ;  of  height  to  thickness  of  piers ; 
of  base  and  coping  to  body  of  pier;  even  the  thickness  of  masonry  courses  (with  due  regard 
for  material)  to  the  general  scale.  A  gradual  curve  over  the  length  of  a  bridge,  with  the  high- 
est point  in  the  centre,  adds  grace  and  prevents  the  optical  illusion  of  a  long  horizontal  line 
appearing  sagged.  Technical  conditions  frequently  necessitate  such  grades,  which  are  either 
regular  curves  or  broken  lines  with  intersections  in  pier  centres.  Much  may  be  done  to  im- 
prove appearance  by  grouping  spans  according  to  importance  and  location,  but  in  such  cases 
the  change  should  be  abrupt  rather  than  gradual ;  as,  for  instance,  a  heavy  pier  may  be 
inserted,  allowing  the  portion  to  one  side  to  act  as  main  structure,  the  other  as  approach 
(Fig.  5,  PI.  XXI).  It  would  also  seem  inappropriate  to  repeat  the  form  of  a  large  span  in  a 
small  one,  thus  producing  the  effect  of  a  miniature  or  model  from  which  the  larger  was 
designed.  The  small  span  should  have  a  character  of  its  own,  to  imply  its  difference  of  pur- 
pose. 

Should  the  number  of  spans  in  a  bridge  be  even,  then  the  centre  pier  should  have  some 
extraordinary  feature  to  mark  its  importance.  When  the  number  is  odd,  the  centre  span 
should  be  longest,  and  if  possible  contain  the  highest  point  of  the  grade  line  (Figs.  3  and  5, 
PI.  XXI). 

In  the  case  of  a  long  viaduct  the  appearance  may  be  greatly  improved  by  a  systematic 
grouping  of  piers  or  spans ;  as  by  inserting  a  masonry  pier  between  a  number  of  metaUic 
towers  or  bents  at  regular  intervals,  or  by  making  spans  alternate  in  length.  This  will  reheve 
monotony  which  would  otherwise  be  a  serious  objection. 

The  most  pleasing  effect  is  produced  by  an  arch,  whether  of  stone  or  metal,  but  is  often 
avoided  on  account  of  expense  (Pis.  XXVI  and  *XXXI).  The  next  form  is  probably  the  in- 
verted arch,  or  suspension  bridge  (Fig.  i,  PI.  XXII),  then  follow  trusses  with  curved  chords 
(Fig.  I,  PI.  XXVIII,  and  Fig.  2,  PI.  XXX).  Still  less  pleasing  in  appearance  are  parallel 
chords  (Figs.  2  and  3,  PI.  XXVII);  and  last  in  order  of  beauty,  irregularly  broken  lines,  as  in 
a  cantilever  (Fig.  6,  PI.  XXII).  The  relative  amount  of  material  for  an  arch  or  suspension 
bridge  is  much  less  than  for  a  truss,  yet  the  cost  of  the  former  is  usually  higher,  for  reasons 
depending  on  manufacture  and  building. 

If  we  wish  to  regard  clearness  of  purpose  as  the  prime  factor  in  determining  what  is 
beautiful,  then  it  is  plain  that  the  suspension  system  should  be  most  popular,  its  principles 
being  plain  to  nearly  every  one.    The  arch  would  rank  next,  and  trusses  last.    It  might  be 


THE  ESTHETIC  DESIGN  OE  BRIDGES. 


421 


added  that  a  beam  or  plate  girder  is  simplest  of  conception,  which  is  true  within  the  limits  to 
which  they  are  applicable.  The  truss,  however,  in  virtue  of  its  internal  complications  loses 
its  resemblance  to  a  beam,  and  is  therefore  less  popular. 

It  is  also  easy  to  understand  why  a  deck  bridge  should  be  more  aesthetic  than  a  through 
bridge.  When  the  supports  are  at  the  top  chord  (where  they  should  be  for  a  truss)  the  struc- 
ture is  in  stable  equilibrium,  and  the  sway  system  can  be  properly  developed,  which  favors 
architectural  perfection.  It  is  to  be  regretted  that  deck  bridges  are  so  limited  in  their  adap- 
tion. 

Ornamentation  if  desirable  should  be  applied  in  accordance  with  the  rules  given  in  the 
articles  on  that  subject. 

439.  Roadway. — The  purpose  of  all  bridges  is  to  carry  loads.  For  railroad  and  highway 
accommodations  this  is  accomplished  over  a  roadway,  for  aqueducts  through  tubes,  etc. 
Hence  the  symbol  of  utility  is  exhibited  by  the  line  carrying  these  loads. 

In  accordance  with  the  foregoing  principles,  it  would  seem  proper  to  develop  this  line  to 
its  full  aestlietic  value.  We  accomplish  this  end  by  adding  weight  in  the  form  of  cornices, 
ornaments,  railings,  etc.,  to  the  line  of  the  roadway.  The  simplest  addition  of  this  kind,  for  a 
metal  bridge,  consists  of  the  floor  system  itself,  and  of  a  plain  coping-stone  for  a  masonry 
structure.  Both  represent  in  themselves  continuous,  heavy  lines,  but  these  may  be  farther 
developed  into  artistic  forms.  As  long  as  we  avoid  extremes,  either  by  supplying  too  much, 
or  by  destroying  the  harmony  with  the  style  of  other  parts  of  the  structure,  a  decided  im- 
provement will  be  realized  by  developing  this  member. 

The  parts  carrying  the  roadway  receive  artistic  form  by  the  addition  of  a  cornice,  the 
character  of  which  depends  upon  the  material  and  style  of  the  structure.  Masonry  is  best 
adapted  to  such  ornamentation,  since  arches  are  always  deck  bridges  and  the  natural  finish  is 
produced  by  the  cornice  and  balustrade. 

It  would  seem  proper  to  omit  cornices  from  through  bridges,  since  there  is  always  a 
strong  tendency  to  cover  the  bottom  chord  to  such  an  extent  as  to  destroy  its  aesthetic 
significance.  This  severe  criticism  may  be  justly  applied  ot  the  Callowhill  Street  Bridge  in 
Philadelphia,  where  both  chords  are  completely  disguised. 

Plate  XVIII  shows  a  number  of  cornices  and  string  courses  to  be  used  in  building?  and 
which  serve  as  suggestions  for  cornices  for  both  stone  and  metal  bridges ;  some  alterations, 
however,  would  be  necessary  to  suit  them  to  the  latter  material.  They  all  require  heavy 
structures  and  would  be  out  of  place  on  slender  designs.  Plates  XIII  to  XVII  give  good 
examples  for  the  various  styles  of  bridges. 

440.  Railings. — For  trusses,  the  only  available  decoration  of  the  roadway  would  seem  to 
be  the  railing.  The  top  chord  of  a  deck  span  may  be  surmounted  by  a  cornice,  but  great  care 
must  be  exercised  not  to  disturb  the  proper  balance.  A  railing  serves  as  a  protection  and 
reminds  the  observer  of  the  intended  purpose  of  a  structure.  Hence  the  lack  of  supplying 
this  member  to  a  public  passage-way  would  invariably  recall  an  insufficiency  of  design. 

In  classifying  the  forms  of  railings  or  balustrades  the  technical  considerations  would  de- 
pend principally  upon  the  materials,  stone,  wood,  or  metal,  while  the  aesthetic  properties  are 
determined  by  the  static  requirements  to  which  the  material  is  made  to  conform.  A  plain 
parapet  wall  is  the  simplest  balustrade  and  is  frequently  applied  to  stone  bridges.  The  first 
motive  to  decoration  is  the  addition  of  a  cover  or  coping,  which  serves  to  protect  the  wall 
against  water.  The  complete  form,  however,  would  necessitate  a  base.  The  artistic  idea  of 
a  railing,  therefore,  embodies  three  parts:  coping,  wall  or  web,  and  base.  These,  though 
derived  originally  from  stone  architecture,  retain  their  aesthetic  value  when  applied  to  wood  or 
metal  designs. 

The  relation  between  the  heights  of  these  parts  to  the  whole  is  important.  There  is  f 
tendency  to  elaborate  the  base  and  coping  beyond  their  proper  proportion.    This  should  be 


422 


MODERN  FRAMED  STRUCTURES. 


carefully  avoided,  as  it  makes  a  bridge  appear  too  light  for  its  purpose.  Naturally,  the  web  o\ 
a  railing  is  most  subject  to  ornamentation,  and  is  usually  divided  into  panels,  which  may  be 
solid  or  otherwise  to  harmonize  with  the  entire  structure. 

Plate  XX  gives  a  number  of  designs  for  railings  which  may  be  used  in  the  following 
manner:  Figs,  i  and  6  are  intended  for  execution  in  brick;  Figs.  2  to  5  in  stone  or  terra 
cotta  ;  Figs.  7  to  10  in  cast-iron.  Fig.  11  may  be  either  cast-  or  wrought-iron  and  is  the 
corresponding  metal  form  to  the  stone  railing  shown  in  Plate  XIV,  Fig.  i.  Figs.  12  to  32  are 
intended  for  wrought-iron,  some  of  which  have  cast  trimmings  and  posts.  Figs.  29  and  31 
represent  cast  frames  with  wrought  fillings.  Other  handsome  designs  may  be  found  in  plates 
XIV  to  XVII. 

Before  closing  the  subject  of  "  .Esthetic  Design  "  there  is  one  other  feature,  not  bearing 
on  design  but  on  execution,  which  ought  to  receive  mention  here.  Nothing  detracts  more  from 
the  general  appearance  of  a  structure  than  an  untidy,  slovenly  surrounding.  Not  infrequently 
quantities  of  refuse  building  material  are  left  on  the  site  of  an  otherwise  handsome  bridge.  A 
little  attention  in  this  direction  would  add  considerable  credit  to  the  parties  concerned. 

COMMENTS  ON  THE  PLATES. 

441.  Plate  XIV. — Fig.  i  may  be  considered  artistic  when  we  allow  for  the  character  of  its 

surroundings,  among  massive  buildings  in  the  city  of  Hamburg.  The  arch  is  plain  and 
heavy,  with  just  enough  detail  to  separate  the  ring  from  the  other  masonry.  The  railing  and 
cornice  are  in  good  proportion,  and  are  well  combined  with  the  coping  of  the  pier.  The 
aesthetic  appearance  might  be  somewhat  improved  by  diminishing  the  cross-section  of  the 
portion  of  the  pier  above  the  springing  of  the  arches  in  some  such  manner  as  in  Fig.  I,  PI.  XV. 

Fig.  2  is  at  fault,  owing  to  the  omission  of  a  coping  at  the  springing  of  the  arches,  as  a 
consequence  of  which  the  latter  have  no  visible  support.  There  is  no  necessity  for  extending 
the  full  section  of  pier  to  the  height  indicated. 

Fig.  3.  The  base  of  the  tall  pier  might  be  modified  as  shown  in  a,  otherwise  the  design 
is  good. 

Fig.  4.  The  tower  above  the  roadway  seems  abnormally  heavy  in  proportion  to  the  pier. 
This  is  caused  by  the  change  of  batter,  which  might  have  been  avoided. 

Fig.  5  is  a  very  neat  design,  though  a  slight  batter  of  piers  and  abutments  would  add  to 
its  beauty. 

Fig.  6  is  open  to  the  same  criticism  as  Fig.  2.  The  coping  at  the  springing  line  should 
have  been  carried  around  the  pier. 

Figs.  7  and  8  are  very  skilfully  treated. 

442.  Plate  XV. — AH  the  figures  of  this  plate,  though  differing  widely  in  character  and 
purpose,  are  good  examples  of  artistic  design.  The  pier  in  Fig.  3  combines  usefulness  with 
a  pleasing  appearance,  and  was  actually  intended  for  a  means  of  defence.  Such  forms  are 
out  of  date,  yet  retain  their  beauty  even  to  the  present  day.  The  towers  in  Fig.  7  are  some- 
what lacking  in  aesthetic  stability.  The  defect  might  easily  be  remedied  by  slightly  inclining 
them  toward  each  other. 

443.  Plate  XVI. — Fig.  i  is  of  a  character  similar  to  Fig.  3,  PI.  XV,  and  would  hardly  be 
applicable  to  this  country.  The  buttress  adjoining  the  arch  in  Fig.  7  might  properly  be  im- 
proved by  a  slight  batter  to  the  left,  with  a  corresponding  widening  of  its  base.  The  other 
examples  on  this  plate  are  hardly  subject  to  any  criticism. 

•  444.  Plate  XVII. — Fig.  i  exhibits  a  very  tasteful  design  for  a  city  bridge.  The  shore 
span  seems  rather  short  as  the  abutment  is  placed  in  the  water,  but  the  conditions  justified 
this  disposition  owing  to  the  previously  existing  quai-walls.    The  centre  of  the  arch  is  properly 


THE  ESTHETIC  DESIGN  OF  BRIDGES. 


423 


offset  by  the  keystone,  and  is  further  emphasized  by  a  similar  ornament  in  the  railing.  The 
recesses  in  the  roadway  at  the  piers  and  abutments  are  very  appropriate  and  add  considerable 
to  the  decoration.  The  cornice  and  railing  over  the  pier,  though  different  fropi  that  of  the 
continuous  roadway,  agree  well  with  the  general  style  of  the  structure.  The  arch  ring  is  well 
developed  and  stands  out  in  relief  from  the  plain  surfaces  of  the  external  segments. 

Fig.  2.  Here  the  line  of  the  supports  is  carried  through  in  the  form  of  a  secondary 
cornice,  actually  dividing  the  bridge  into  two  stories,  which  idea  is  artistically  embodied  in 
the  centre  pier.  The  entire  design  is  plain  and  complies  well  with  the  requirements  of  a  rail- 
road bridge. 

Fig.  3  is  a  more  elaborate  structure  for  a  railroad,  and  is  well  suited  to  its  location. 

Fig.  4  illustrates  the  utility  of  brick  in  erecting  an  inexpensive  bridge.  All  the  decora- 
tions are  of  this  material  produced  by  color-effect  and  otherwise.  The  pier  would  be  more 
graceful  if  the  portion  above  the  lower  coping  were  not  quite  so  wide. 

445.  Plates  XVIII,  XIX,  and  XX  are  considered  in  articles  pertaining  to  roadway. 

446.  Plate  XXI. — Fig.  i  owes  its  pleasing  apppearance  solely  to  its  form.  The  struc- 
ture is  very  plain  otherwise,  but  suits  its  purpose  admirably. 

Fig.  2  is  of  similar  character,  but  adapted  to  railway  traffic.  The  dimensions  of  the  arch 
increase  toward  the  springing,  which  illustrates,  architecturally,  the  technical  requirements. 

Fig.  3.  This  immense  structure  justly  demands  supremacy  over  its  surroundings.  The 
graceful  lines  of  the  cables  stand  out  in  bold  relief  against  the  sky,  thus  bringing  about  per- 
fect harmony  with  the  landscape,  which  is  characterized  by  the  masts  and  rigging  of  thou- 
sands of  vessels  in  the  harbor.  The  bridge  is  strictly  plain  and  owes  its  beauty  to  its  lorm. 
The  towers  have  the  appearance  of  being  unfinished,  and  should  be  capped  with  an  appro- 
priate design. 

Fig.  4  is  a  plain  though  very  effective  design  for  a  viaduct.  It  might  not  be  as  economi- 
cal as  a  plate  girder,  but  the  appearance  is  worth  the  difference.  It  is  a  fine  example  of  artis- 
tic construction  in  wrought-iron. 

Fig.  5  represents  one  of  the  few  bridges  that  may  be  called  a  model.  Every  detail  dis- 
closes artistic  tact. 

447.  Plate  XXII. — As  a  general  criticism,  all  these  designs  lack  symmetry  except  Nos. 
I,  2,  and  8,  which  defect  is  not  excusable,  as  may  be  seen  from  PI.  XXIII.  No.  i  is  a  correct 
solution  of  the  problem,  and  presents  a  very  graceful  appearance. 

No.  2  is  wanting  in  aesthetic  stability,  as  the  thin  cables  of  the  approaches  do  not  counter- 
balance the  mass  of  the  main  span. 

No.  3  conveys  the  idea  of  extreme  economy.  Such  a  structure  would  not  be  very  credit- 
able to  a  wealthy  city. 

No.  4  is  a  succession  of  arches  which  bear  no  artistic  relation  to  each  other.  The  heavy 
pier  in  the  bridge  centre  is  not  called  for,  since  the  long  span  is  really  more  important.  The 
change  of  grade  was  unnecessary,  as  shown  by  the  other  designs. 

Nos.  5  and  9.  If  the  left  half  had  been  repeated  on  the  right-hand  side  of  the  centre  pier, 
both  designs  would  have  been  acceptable.  Of  course  this  would  convert  the  probable  canti- 
lever into  two  braced  arches  and  two  cantilever-arms. 

No.  6  would  seem  like  a  monstrosity  in  a  city  like  New  York.  Such  a  bridge  would  be 
more  suitable  to  the  wilds  of  the  far  West. 

No.  7  contains  such  a  variety  of  arches  that  the  common  mind  might  wonder  whether 
there  could  be  any  circumstance  where  arch  construction  is  not  possible.  Of  course  the  de- 
signer probably  had  some  reason  for  choosing  this  disposition,  yet  the  manner  of  solving  the 
problem  is  justly  open  to  criticism. 

No.  8  is  too  heavy  and  would  not  harmonize  with  the  surroundings. 

No.  10  would  represent  a  fair  design,  though  the  centre  span  might  have  been  increased 
about  50  feet.    Even  then  the  river  spans  seem  rather  insufficient. 


424 


MODERN  FRAMED  STRUCTURES. 


448.  Plate  XXIII. — The  four  designs  for  Harlem  River  or  Washington  Bridge  illustrate 
very  clearly  how  the  unsymnietrical  forms  on  PI.  XXII  might  have  been  avoided.  The  draw- 
ing by  Mr,  C.  C.  Schneider  is  a  model  in  every  respect.  The  metal  arches  are  highly  artistic, 
yet  plain,  and  appeal  to  the  eye  as  the  important  members  of  the  bridge.  The  more  orna- 
mental roadway  is  massive,  but  in  very  good  proportion  to  the  arches.  The  masonry  ap- 
proaches exhibit  a  distinct  character,  peculiar  in  themselves  The  style  of  the  shore  piers  is 
exactly  repeated  in  the  river  pier.  The  coping  which  marks  the  springing  of  the  masonry 
arches  is  carried  through  the  entire  stone  work.  The  railing  and  cornice  are  uniform  over  the 
bridge,  but  are  emphasized  in  the  main  structure  by  the  secondary  arch  system  below.  The 
harmony  between  the  various  parts  is  admirably  preserved. 

The  design  by  Mr.  W.  Hildenbrand  is  less  pleasing  in  effect.  The  arches  of  the 
approaches  resemble  too  much  the  secondary  system  of  the  roadway,  which  latter  appears 
rather  heavy  compared  with  the  main  arches  which  carry  them.  The  manner  of  bracing  the 
vertical  columns  destroys  to  a  great  extent  the  effective  character  of  the  braced  arches  which 
stand  out  so  clearly  in  Mr.  Schneider's  design. 

The  defects  in  the  contract  drawing  are  nearly  all  remedied  in  the  bridge  as  built,  and  are 
so  plainly  visible  that  nothing  more  need  be  added. 

449.  Plate  XXIV. — Figs,  i,  2,  and  3  are  very  neat  designs,  each  suited  admirably  to  its 
purpose.  The  beauty  is  due  entirely  to  the  general  form,  and  decorations  would  be  quite 
unnecessary.    The  lack  of  symmetry  is  clearly  justified  by  the  profile. 

The  St.  Louis  and  Washington  bridges  are  considered  elsewhere. 

450.  Plate  XXV. — Fig.  i.  This  most  beautiful  structure  well  deserves  mention  here,  and 
illustrates  the  extent  to  which  aesthetic  design  may  be  applied  to  bridges.  It  is  hoped 
that  we  may  soon  be  able  to  include  similar  artistic  feats  among  American  productions. 

Fig.  2.  The  Salzburg  Bridge,  together  with  its  landscape,  exhibits  a  very  picturesque 
appearance.  The  only  defect  is  in  the  peculiar  truss  ornamentation,  which  partially  disguises 
the  technical  outline  of  the  superstructure. 

451.  Plate  XXVI. — Fig.  l  is  a  railroad-  and  highway-bridge  combined,  which  accounts 
for  the  heavy  form,  though  the  design  is  in  good  keeping  with  tlie  surroundings. 

Fig.  2.  The  iron  highway  bridge  in  Basel  is  a  very  pretty  structure.  The  masonry  is 
rather  old-fashioned,  but  this  is  required  by  the  character  of  the  locality.  The  river  piers  are 
still  in  want  of  the  necessary  statuary. 

Fig.  3  shows  what  results  may  be  obtained  without  resorting  to  ornamentation.  The 
bridge  represents  a  most  effectual  piece  of  work,  comparatively  cheap  and  to  the  purpose. 

452.  Plate  XXVII. — Fig.  i  illustrates  a  type  of  bridge  which  should  be  more  frequently 
seen  in  the  parks  of  our  large  cities.  It  is  a  very  handsome  design,  with  the  exception  of  the 
railing.  One  of  the  patterns  on  Plate  XX,  as  Figs.  21,  25,  or  30,  would  have  been  more  in 
place. 

Fig.  2  is  in  itself  far  from  being  a  beautiful  structure,  though  it  is  in  artistic  balance  with 
the  rugged  landscape.    Nothing  elaborate  is  called  for,  hence  the  design  answers  its  purpose. 

Fig.  3  represents  the  earliest  form  of  metal  bridges,  with  piers  adapted  to  defensive 
purposes.  Even  at  the  present  time  these  might  prove  useful.  The  general  disposition  is 
nevertheless  in  conformity  with  aesthetics.  The  centre  pier  is  developed  to  the  proper  degree, 
and  the  shore  piers,  or  abutments,  are  offset  by  the  fortified  towers,  which  include  the  end 
portals  and  toll  offices.  The  lattice  trusses  rank  among  the  least  artistic,  but  in  this  case,  with 
its  many  systems,  it  approaches  closely  the  beam  or  tubular  form. 

453.  Plate  XXVIII. — -Fig.  l  is  a  modern  example  of  the  fortified  type  of  bridges.  The 
abutment  is  a  perfect  fort,  to  which  the  heavy  trusses  are  well  suited. 

Fig.  2  is  a  very  graceful  and  highly  ornamental  bridge,  of  which  the  city  of  Mayence  may 
well  be  proud. 


THE  ESTHETIC  DESIGN  OF  BRIDGES. 


425 


Fig.  3.  This  is  one  of  the  earliest  forms  of  metal  arches,  and  was  used  largely  as  a  pattern 
for  the  Eads  Bridge  in  St.  Louis.  It  is  a  plain  yet  effective  design  and  highly  creditable  to 
its  engineer. 

454.  Plate  XXIX.— Fig.  i.  The  Rialto  Bridge,  over  the  Grand  Canal,  in  Venice,  is  his- 
torical in  bridge  architecture.  It  is  the  peculiar  character  of  the  adjoining  dwellings  that  lends 
beauty  to  this  monument  to  early  engineering.  The  same  design  repeated  elsewhere  would 
undoubtedly  prove  an  artistic  failure. 

Fig.  2  is  a  beautiful  structure  and  very  creditable  to  the  city  of  St.  Louis.  It  is  beyond 
question  one  of  the  neatest  suspension  bridges  in  this  country,  and  would  bear  repetition  else- 
where. 

Fig.  3  represents  the  model  bridge  of  Philadelphia.  Similar  structures  erected  at  Market 
Street,  Girard  Avenue,  and  Callowhill  Street,  would  have  been  more  in  place  than  the  present 
crossings  of  the  Schuylkill  at  these  points.  The  Girard  Avenue  Bridge  is  very  handsome,  but 
does  not  compare  with  the  one  at  Chestnut  Street.  Instead  of  the  abrupt  change  of  grade  in 
the  centre  of  this  bridge,  a  gradual  curve  might  have  been  chosen. 

455.  Plate  XXX. — Fig.  i.  The  portal  of  the  New  Hamburg  Bridge  is  one  of  the  finest 
pieces  of  engineering  architecture  in  existence.  The  superstructure,  which  is  an  exact  copy  of 
that  of  the  old  bridge  shown  in  Fig.  2,  would  hardly  be  duplicated  elsewhere,  though  it  looks 
well.   The  fact  that  two  structures  of  this  design  were  placed  side  by  side  is  to  be  deplored. 

456.  Plate  XXXI. — Fig.  i  of  this  plate  has  received  mention  in  speaking  of  Plate  XXIV. 
The  St.  Louis  Bridge  with  its  world-wide  reputation  scarcely  needs  any  comment  here; 

it  is  added  as  a  model  of  aesthetic  design, 

456a,  Plate  XXXIa. —  Fig.  i  *  is  a  fTne  view  of  the  Tower  Bridge  recently  constructed 
across  the  Thames  at  London,  near  the  Tower  of  London,  from  which  it  takes  its  name.  This 
design  is  the  result  of  some  twenty  years'  continuous  study  and  discussion,  and  although  it 
spans  a  total  opening  of  only  about  800  feet,  it  has  been  constructed  at  a  cost  of  over 
$4,000,000.  A  large  part  of  this  cost  has  been  incurred  purely  for  the  sake  of  aesthetic  effect. 
The  four  elegant  stone  towers  simply  serve  the  purpose  of  enclosing  and  covering  from  view 
four  pairs  of  steel  towers  which  carry  the  loads  imposed  by  the  suspension  cables.  The  two 
side  openings  are  270  feet  each,  and  the  centre  opening  200  feet.  The  roadway  of  the  central 
span  is  arranged  to  open  on  the  bascule  principle,  and  while  it  is  open  passengers  are  transferred 
to  the  tops  of  the  towers  by  elevators  and  cross  over  on  two  independent  footways,  each  twelve 
feet  wide,  at  a  height  of  141  feet  above  high  tide.  This  bridge  was  opened  for  service  in 
June,  1894,  and  has  continued  in  very  successful  and  satisfactory  operation  to  the  present 
writing  (May,  1895).  The  design  of  this  bridge  is  due  to  J.  W.  Barry,  Engineer,  and  Horace 
Jones,  Architect,  both  of  whom  were  employed  by  the  city  of  London  to  prepare  and  execute 
the  design.  It  forms  the  latest  and  best  illustration  of  the  proper  course  a  corporation  should 
pursue  in  securing  the  best  results  where  an  engineering  design  is  expected  to  have  a  pleasing 
architectural  effect.  Neither  the  engineer  nor  the  architect  alone,  however  competent  in  his 
own  field,  is  equal  to  the  successful  execution  of  such  a  project.  The  result  has  more  than 
justified  the  action  of  the  corporation  of  the  city  of  London  in  this  matter. 

Fig.  2  f  illustrates  the  Black  Friars  Road  Bridge  across  the  Thames  at  London,  which  was 
finished  in  1869.  Mr.  Joseph  Cubitt  was  the  engineer  engaged  by  the  city  corporation.  The 
piers  are  ornate,  but  in  excellent  taste,  the  cylindrical  column  being  composed  of  polished  red 
granite,  and  the  pedaments  and  capitals  of  Portland  stone.  The  external  wrought-iron  ribs 
are  covered  with  ornate  cast-iron  facia,  and  the  slightly  curved  roadway  is  guarded  on  either 
side  with  a  parapet  of  Gothic  design,  three  feet  nine  inches  high.  This  bridge  is  an  excellent 
example  of  simplicity  and  good  taste,  obtained  at  a  moderate  cost. 


*  See  Engineering,  Jan.  4,  1895. 


f  See  Engineering,  Feb.  8,  1895. 


426 


MODERN  FRAMED  STRUCTURES. 


4566.  Plate  XXXI^.  —  Fig.  i  *  of  this  plate  shows  the  Charing  Cross  railway  bridge, 
completed  in  1869  and  widened  in  1884.  But  little  attempt  has  been  made  in  this  bridge  to 
produce  a  pleasing  architectural  effect,  and  the  pile-like  character  of  the  piers  gives  to  the 
bridge  the  air  of  a  temporary  structure.  This  bridge  was  built  by  the  railroad  companies  who 
use  it,  and  it  well  illustrates  the  difference  between  a  good  and  a  poor  architectural  design. 
Even  such  attempts  as  have  been  made  at  ornamentation  on  the  brick  piers  appear  strained 
and  out  of  harmony  with  the  rest  of  the  design. 

In  Fig.  2  f  we  have  a  view  of  the  Victoria  Road  Bridge  across  the  Thames,  which  was 
completed  in  1858,  after  a  design  by  Mr.  Thomas  Page.  The  total  length  of  this  bridge  is  700 
feet,  the  central  opening  being  333  feet.  Both  the  abutments  and  the  piers  are  enclosed  in 
stonework,  giving  a  very  pleasing  architectural  effect.  The  loads  are  really  carried  inside  of 
these  stone  shells  on  iron  columns  and  concrete  masonry.  As  the  anchorage  abutments  are 
on  the  river  banks,  and  the  two  piers  are  placed  in  the  river,  the  entire  structure  rests  over 
the  water,  and  nothing  has  been  lost  by  having  the  approaches  extend  over  the  land.  It 
offers  less  obstruction  to  river  trafific  than  an  arch  bridge  would,  and  it  has  a  more  pleasin;^ 
appearance,  especially  by  way  of  contrast. 

456c.  Plate  XXXIc. — This  plate  contains  an  additional  view  of  the  Tower  Bridge 
London,  shown  in  Fig.  i,  Plate  XXXI^?. 

456d.  Plate  XXXI^/. — This  plate  gives  a  view  of  the  famous  suspension  bridge  at  Buda- 
Pesth,  which  is  frequently  called  the  handsomest  bridge  in  the  world. 

Note. — The  author  is  greatly  indebted  to  the  following  gentlemen,  who  have  so  kindly  assisted  in  procuring 
.desirable  illustrations  : 

Messrs.  Wm.  R.  Hutton  and  Leo  von  Rosenberg,  for  the  use  of  Plates  XIII,  XIX,  XXII,  XXIII,  and  XXIV  from 
the  monograph  on  the  Washington  Bridge  ; 

Mr.  John  C.  Trautwine,  Jr.,  for  view  of  Chestnut  Street  Bridge,  Philadelphia; 

Mr.  Carl  Gayler,  M.  Am.  Soc.  C.  E.,  Designer,  and  the  King  Bridge  Co.  Contractor,  for  view  of  Grand  Avenue 
Bridge,  St.  Louis ; 

The  Cosmopolitan  Magazine,  for  the  plates  of  Fig.  3,  Plate  XXI  ;  Fig.  i,  Plate  XXIX  ;  and  Fig.  i,  Plate  XXXI  ; 

Prof.  C.  L.  Crandall,  of  Cornell  University,  for  the  use  of  Plate  XVIII,  and  a  number  of  photographs  from  the 
magnificent  collection  belonging  to  that  institution  ; 

Mr.  James  Dredge,  Editor  of  Engineering,  for  the  four  views  of  London  bridges  shown  in  Plates  XXXI«  and 
XXXI^. 


*  See  Engineering,  Feb.  22,  1895. 


f  See  Engineering,  April  5,  1895. 


PLATE  XIII. 


THE  ALBERTYPE  CO.,  N.  Y. 


CENTRAL  PIER. 

Washington  Bridge. 


Plate  XVI. 


Fig.  2. 


Portal  of  Lahn  Bridgk,  Nassau. 


Portals  of  Danube  Bridges,  Vienna. 
Fig.  4.       ^  Fig.  5. 


Fig.  3. 


Street-crossing  in  Strassburg. 


Carola  Bridge 
near  Schandau. 


CORNICES. 


Plate  XVIII. 


Plate  XXI. 


Vie.  I.  — KlRClIIiNKELD   BRIDGE,   IN   BERN,  Sw  TI /.l-.KI.AND. 


Fig.  2. — SCHWAR/.WASSER  VlADUCr,  SWITZERLANI'. 


Fu;.  4. — Viaduct  over  tiik  Rivi.r  Repiro,  Brazil. 


Fig.  5. — Albert  Bridge,  in  Dresden. 


Plate  XXV. 


Fig.  2. — Metal  Bridge  over  Inn  River  in  Salzburg,  Tyrol. 


Plate  XXVI. 


Plate  XXVII. 


Fir,.  I.  — Bridge  over  Canal,  Belle  Isle  Park,  Detroit. 


Fk;.  2.  — HiciiiwAY  Bridge  over  VVeiira  River,  Weiii;,  Haden. 


Fig.  h. — Bridge  o\er  hie  Ri:ine,  Cologne,  Gicigmanv. 


Plate  XXVIII. 


Flatf.  XXIX. 


Plate  XXX. 


Fig.  2. — Old  Bkidgk  at  Hamburg. 


Plate  XXXT. 


Plate  XXXhr. 


Fig.  2.— BLACKFRIARS  ROAD  BRIDGE.  LONDON. 


Plate  XXXU. 


Fig.  2.— the  VICTORIA  BRlD(iE,  LONDON. 


Plate  XXXIr. 


TOWER  BRIDGE,  LONDON, 


Plate  XXXU. 


Buda-Pesth  Siisi'KNsioN  Hridge. 


STAND-PIPES  AND  ELEVATED  TANKS. 


CHAPTER  XXVII. 
STAND-PIPES  AND  ELEVATED  TANKS. 


457-  Use  of  Water  Towers. — Where  high  ground  is  not  available  for  service  reservoirs 
as  a  part  of  a  city  water  supply  system,  they  are  replaced  by  the  storage  of  a  small  quantity 
of  water  at  a  high  elevation,  in  a  steel  tank  or  "stand-pipe."  These  serve  to  relieve  the  pipe 
system  of  excessive  "  water-rams,"  and  to  equalize  the  irregularities  of  pumping  and  using. 
These  reservoirs  may  be  large  enough,  in  small  cities  and  towns,  to  supply  the  night  service, 
and  also  to  feed  three  or  four  fire  streams  for  an  hour,  or  until  the  pumps  can  be  started. 
In  the  case  of  a  stand-pipe  it  is  only  the  water  in  the  upper  portion  which  is  available  for  fire  ser- 
vice. It  is  on  this  account  that  tanks  of  larger  diameter,  elevated  upon  steel  or  masonry 
towers,  are  always  of  more  value  than  stand-pipes  of  the  same  total  cost.  The  stand-pipes 
are,  however,  much  more  common.  From  the  great  number  of  failures  of  these  in  all  parts 
of  the  country,  in  the  last  few  years,  it  is  evident  that  they  have  been  very  poorly  designed. 

These  two  kinds  of  structures  will  be  discussed  separately. 


Stand-pipes. 

458.  Dimensions. — In  order  to  decide  upon  the  dimensions  of  a  stand-pipe  it  is  necessary 
to  determine  the  storage  capacity  required  above  a  given  plane.  Suppose  this  plane  of  least 
effective  elevation  of  water  to  be  100  feet.  Let  it  be  required  to  compute  the  storage 
capacity  needed  to  serve  a  given  population  through  the  night,  when  the  pumps  are  operated 
only  in  the  daytime,  counting  the  night  service  as  somewhat  less  than  one  half  the  day 
service.  Allowing  60  gallons  each  per  day  for  the  entire  population,  we  have,  as  tlie  height 
through  which  the  tvatcr  will  be  draxvn  dozvn  at  night,  for  stand-pipes  of  different  diameters, 
and  for  towers  of  different  sizes, 

h  =  -~jr-  for  12  hours'  pumping  ; 

h  =  ~-  "  10    "  " 

SP    "    8    "  « 
^  =  d^ 

where  h  =  height  in  feet  through  which  the  water  is  drawn  down  at  night ; 
/*=  number  of  population  ; 
d—  diameter  of  stand-pipe  in  feet. 


(I) 


428 


MODERN  FRAMED  STRUCTURES. 


Thus  for  a  town  of  8000  inhabitants,  with  a  stand-pipe  20  feet  in  diameter,  the  water 
would  be  lowered  50  feet  at  night  for  10  hours'  pumping. 

Again,  let  it  be  required  to  find  the  capacity  necessary  to  supply  four  fire  streams,  each 
discharging  200  gallons  per  minute  (i-inch  smooth  nozzles  70  feet  high.  Freeman),  for  a  period 
of  one  hour.    We  now  have,  as  the  number  of  fire-stream  hours  which  can  be  supplied, 

n  —  o.ooo^hd"  (streams  of  200  gallons  each  per  minute);  ) 
n  =  0.0004/id''  (streams  of  250  gallons  each  per  minute);  I 

where  n  =  number  of  fire-stream-hours  ; 

//  =  height  through  which  the  water  is  drawn  down  ; 
d  =  diameter  of  stand-pipe  in  feet. 

From  equations  (i)  and  (2)  the  capacity  of  the  stand-pipe  above  a  given  plane  can  be 
found.    This  determines  its  height  when  the  diameter  has  been  fixed. 

The  height  should  never  be  more  than  ten  times  the  diameter,  and  preferably  not  more 
than  eight  times  the  diameter. 

459.  Character  and  Thickness  of  Metal. — The  material  of  which  the  plates  are  made 
should  be  mild  steel  having  an  ultimate  strength  of  55,000  to  64,000  lbs.  per  square  inch.  In 
the  markets  this  is  known  as  "  sheet  steel,"  or  "  boiler  steel."  There  is  a  cheaper  grade  of 
steel  plates  on  the  market  known  as  "  tank  steel."  This  is  apt  to  be  hard  and  brittle  and 
shoidd  never  be  allowed  in  any  part  of  the  structure.  Most  of  the  failures  in  stand-pipes  can  be 
traced  to  this  one  cause.*  And  yet  most  of  the  stand-pipes  in  this  country  have  been  built 
of  this  material.    (See  Art.  468  for  specification  for  material.) 

If  sufficient  precaution  is  taken  to  insure  obtaining  the  right  kind  of  material  in  the  plates 
then  the  thickness  can  be  determined  by  the  following  considerations : 

Allowing  that  the  double-riveted  vertical  joints  have  an  ultimate  strength  of  60  per  cent 
of  the  gross  section,  and  counting  the  tensile  strength  of  the  material  at  60,000  lbs.  per  square 
inch,  the  actual  strength  of  the  vertical  joint  is  36,000  lbs.  per  square  inch  of  gross  section. 
Taking  a  factor  of  safety  of  four,  we  may  allow  a  tensile  stress  of  9000  lbs.  per  square  inch  on 
the  gross  section  or  of  15,000  lbs.  per  square  inch  on  the  net  section. 

The  tensile  stress  in  the  shell,  per  vertical  inch,  is  7"  =  />r,  where  p  is  the  fluid  pressure 

per  square  inch,  and  r  is  the  radius  of  the  cylinder  in  inches.    But  p  —  ^^ll^  —  0.434^,  where 

144 

2r 

h  is  the  head  in  feet.  Taking  =  —  as  the  diameter  in  feet,  we  have,  for  the  thickness  of 
the  shell  required  at  any  depth, 


t  =         =  Q).cxx>ihd  (nearly)  (3) 

where  t  is  in  inches  and  h  and  d  are  in  feet. 

The  least  thickness  used  should  not  be  less  than  one  fourth  of  an  inch. 

Fig.  423  shows  graphically  the  thickness  to  use  for  different  values  of  hd,  varying  by 
inch.    It  is  not  desirable  to  specify  thicknesses  varying  by  thirty-seconds  of  an  inch.  This 
diagram  makes  some  allowance  for  imperfect  workmanship  and  a  poorer  grade  of  material  in 
the  very  thick  plates.    It  is  not  possible  to  rivet  up  thick  plates  without  considerable  internal 
stress. 


*  The  writer  examined  and  tested  the  broken  sheets  from  a  stand-pipe  which  burst  under  a  static  pressure  which 
produced  a  tensile  stress  of  only  one  fifth  of  the  ultimate  strength  of  the  material  and  found  them  so  brittle  that  they 
aould  not  be  straightened  in  the  rolls  without  breaking. 


STAND-PIPES  AND  ELEVATED  TANKS. 


429 


hd 


In  Fig.  423,  the  thicknesses  are  shown  to  actual  scale.    It  is  not  wise  to  try  to  use  plates 

thicker  than  one  inch.  If  this  does  not  give  the  capacity 
required  it  would  be  better  to  duplicate  the  plant  than  to 
try  to  use  thicker  plates. 

460.  Wind  Moment  and  Anchorage. — The  over- 
turning moment  due  to  wind  pressure  may  be  taken  as  40 
lbs.  per  square  foot  on  one  half  the  diametral  area  into  one 
half  the  height  if  a  stand-pipe,  or  into  the  height  of  the 
centre  of  pressure  if  an  elevated  tank. 

The  moment  of  stability  is  the  weight  of  the  empty 
tank  into  its  radius  if  there  are  no  brackets,  or  into  the  per- 
pendicular distance  to  a  line  joining  two  adjacent  anchorage 
points  when  brakets  are  employed.  If  this  is  not  equal  to 
the  overturning  moment,  the  remainder  must  be  provided 
for  by  anchorage  rods  extending  into  the  masonry. 

In  Fig.  424,  let  AB  .  .  .  /^represent  the  anchorages  of 
the  brackets  of  a  stand-pipe  whose  centre  is  O.  If  we 
assume  the  wind  to  be  in  the  direction  ON,  then  the 
brackets  A  and  D  act  with  arms  one  half  those  of  B  and  C. 
The  pull  on  the  rods  at  A  and  D  will  also  be  one  half  that 
at  B  and  C,  hence  the  moments  of  resistance  of  the  brackets 
A  and  D  will  be  one  fourth  those  at  B  and  C. 


hd=4000 


Thickness  of  Plates 

Fig.  423. 


Taking  moments  about  the  line  EF, 

Let  Af  —  overturning  moment ; 

M„  —  moment  of  resistance  of  dead  weight  =  IV  X  dist.  OH  - 
P  —  pull  on  anchorage  rods  at  B  and  C;  then  we  have 


M=  WA-\-  4PA  +PA    or  P=z 


M  -  WA 


(4) 


If  the  wind  had  been  in  the  direction  BE,  then  the  anchor  rods  at  B  would  have  acted 


MODERN  FRAMED  STRUCTURES. 


with  the  arm  BE,  and  those  at  A  and  C  with  the  arms  AF  ^wA  CD.  Let  the  distance  OK  — 
a  ;  then  CD  =  AF  =  2a,  and  BE  —  4a. 

The  amount  of  the  Hft  at  A  and  C  is  to  that  at  B  as  their  relative  distances  from  the 
diametral  line  mn  or  as  a  is  to  2a. 

Let      =  pull  on  anchorage  at  B  ; 


Then  we  have 


But  since     =  -L,  we  obtain 
2 


a  500 
Therefore 


M  -2Wa 


6a 


(5) 

(6) 


«  =  0.58^. 

M-  U16WA 


SA^A 


(7) 


By  comparing  this  with  eq.  (4)  it  will  be  seen  that  /*,  is  greater  than  P  when  the  over- 
turning moment  is  greater  than  about  twice  the  moment  of  stability  IVA,  otherwise  F  is  the 
greater. 

Having  found  P  or  /*, ,  whichever  is  the  greater,  this  fixes  the  depth  of  the  masonry 
foundation,  the  size  of  the  anchor  rods,  and  the  strength  of  the  brackets.  The  masonry  must 
be  deep  enougli  to  supply  the  necessary  dead  weight ;  or  that  part  of  it  which  can  be  supposed 
to  rest  on  the  anchor  plate,  or  to  be  lifted  by  this  plate,  must  be  equal  to  P.  The  weight  of 
a  cubic  foot  of  masonry  may  be  taken  at  150  lbs. 

The  size  of  the  anchor  rods  must  be  taken  as  the  size  at  the  base  of  the  screw  threads  if 
the}'  are  not  upset  at  the  ends.  The  working  stress  on  these  may  be  taken  at  15,000  lbs.  per 
square  inch  of  net  section. 

461.  The  Anchorage  Brackets.^ — It  is  a 

very  common  practice,  in  anchoring  down  pedes- 
tals, to  attach  the  anchor  bolts  to  the  outstanding 
legs  of  angle  irons  which  are  riveted  to  the  feet 
of  the  posts.  This  is  a  very  poor  attachment,  as 
at  most  two  or  three  rivets  are  required  to  take 
the  whole  pull  of  the  anchor  bolts.  A  better 
arrangement  for  a  bracket  is  that  shown  in  Fig. 
425. 

The  bracket  is  15  feet  high  and  has  a  base  of 
8  feet  to  anchor  rods.  It  is  curved  slightly  for 
appearance,  and  is  composed  of  a  solid  plate  with 
double  angles  on  all  three  sides.  The  pull  from 
the  anchor  bolts  is  transmitted  through  a  suf- 
ficient number  of  rivets  to  the  web  of  the  bracket, 
and  thence  by  shear  and  bending  moment  to  the 
side  of  the  stand-pipe,  which  is  reinforced  on  the 
inside,  at  the  top  of  the  bracket,  by  a  heavy  angle 
iron.  This  takes  the  pull  or  thrust  coming  from 
the  outer  flange  of  the  bracket  which  is  concen- 
trated at  the  upper  end.* 


Fig.  425. 


*  This  form  of  bracket  was  used  on  the  stand-pipe  at  Jefferson  City,  Mo.,  which  is  shown  in  Fig.  430. 


STAND-PIPES  AND  ELEVATED  TANKS. 


431 


0  11 

1  0 

0  1 

1  0 

0  1 

1  0 

of 

1  ^ 

0  1 

1  0 

0  1 

1  0 

0 

1  0 

0 

1  0 

0  1 

1  0 

0  i 

1  ^ 

0 

[Jo 

l3h 


1 


For  large  brackets  stiffening  angles  should  be  placed  diagonally  across  the  web,  as  shown 
in  Fig.  425. 

In  case  no  brackets  are  used  the  anchor  bolts 
should  be  attached  directly  to  the  bottom  ring  of  the 
stand-pipe  by  means  of  long  angle  irons,  as  shown  in 
Fig.  426. 

The  anchor  bolts  must  here  be  kept  as  close  to 
the  side  of  the  stand-pipe  as  possible. 

The  brackets  are  preferable,  however,  as  they 
serve  also  to  distribute  the  dead  weight  over  a  larger 
area  of  foundation.  Furthermore,  they  add  greatly  to 
the  general  appearance  of  the  structure.  (See  Fig,  430.) 

The  use  of  cast-iron  lugs,  of  short  vertical  dimen- 
sions, riveted  to  the  sides  of  the  stand-pipe,  to  which 
the  anchor  bolts  are  attached,  cannot  be  too  severely 
condemned. 

462.  Details  of  Construction. — This  not  being  a  work  on  foundations  or  on  water-works 
construction,  the  questions  pertaining  to  character  and  sufificiency  of  the  foundation;  inlet  and 
outlet  pipe  ;  cut-off  valves  and  their  operation  by  hand  or  by  electricity  from  the  pumping 
station;  float  indicators,  electric  and  otherwise;  man-holes,  stairway,  flushing-out  pipe,  etc.,  are 
here  omitted  ;  also  all  discussion  of  outer  masonry  structural  housing  and  the  design  of  the 
same,  and  whether  or  not  it  is  necessary  to  enclose  the  stand-pipe  in  any  manner.  It  is 
assumed,  however,  in  this  chapter  that  the  tower  is  not  enclosed. 


Riveting.- 


-The  subject  of  riveting  is  discussed  in  Chapter  XVIII. 


Although  lap  joints 

are  almost  universal  in  these  structures,  the  diflficulty  of  making  a  water-tight  joint  where 
three  plates  come  together  makes  some  other  arrangement  desirable.    The  strength  of  a  lap 


Fig.  427. 


joint  also  is  much  less  than  that  of  a  double-strap  butt  joint.  Since  the  horizontal  joints  are 
not  stressed  by  the  water  pressure,  they  may  be  lapped  and  single  riveted.  The  vertical 
joints  should  be  made  with  double  butt  straps,  as  shown  in  Fig.  427.  Here  the  vertical  joints 
have  double  butt  straps  with  bevelled  calking  edges  on  all  four  sides  of  the  outer  strap.  The 
inner  strap  is  not  calked.  The  straps  should  be  not  less  than  \  in.  thick,  nor  thinner  than  one- 
half  the  thickness  of  the  plate. 


432 


MODERN  FRAMED  STRUCTURES. 


The  proportions  of  diameter  and  pitch  of  rivets  to  thickness  of  plate  are  given  in  the 
table  below,  which  has  been  compiled  from  the  Watertown  Arsenal  experiments  on  riveted 
joints. 

RIVETED  JOINTS  FOR  STAND-PIPES. 


ls.lnu  01  Joint  on 

I  ntcKncss 

n 

Pitch 

of  Rivets 

Distance 
between 
Pitch-lines. 

Distance 
of  Pitch-line 

Percentage 

fJl   1  ULcil  oLrcn^iit 

Holes. 

Vertical  Seams. 

of  Plate, 

of  Rivet. 

Centre 

from 

ot  Plate 

to  Centre. 

Edge  of  Plate. 

developed. 

inch 

inch 

inch 

inch 

inch 

Single-riveted  lap. . . . 

i 

H 

I 

50 

B 

T5 

If 

li 

Double-  "  " 

i 

8 

■^8 

2i 

lA 

<l                  it  (C 

< (          ((          ( < 
<<           t(  <c 

1 

7 

i 

2f 
24 

2i 
2i 

25 

li 
If 
14 

60 

< 

►  Punched 

"      "     butt '. '. . 

1 

1 

2' 
•^T 

2k 

It 

(1             <l  ti 

7 

'S 

2| 

2h 

If 

«<             ((  <( 

1 1 

Ttr 

2| 

2t 

If 

<<            it  <( 
((            <<  (( 

f 

1  3 

3 
3 

2i 

2i 

1* 
I| 

70 

«C               ((  (( 

3 

2i 

2 

C(               ii  t< 

IB 

li 

T  1 

3 

2* 

2 

-Drilled 

ft           (f  <t 

I 

3i 

2| 

2i 

Triple-  " 

I 

4 

3 

2i 

75 

( 


It  will  be  observed  that  the  butt  joint  is  much  more  efificient  than  the  lap  joint,  even 
when  both  are  double  riveted.  When  the  plates  are  more  than  f  in.  thick  they  should  be 
drilled  and  not  punched,  as  the  cold  flowing  of  the  metal  in  punching  thick  plates  injures  it 
for  a  considerable  distance  around  the  holes. 

Plates  thicker  than  i  in.  should  not  be  used  if  it  can  be  avoided.  Since  all  the  riveting 
is  done  by  hand  in  the  field,  the  rivets  should  be  of  iron. 

Calking. — All  calking  edges  should  be  bevelled  on  a  planer,  and  the  calking  should  always 
be  done  with  a  round-nosed  tool,  as  shown  in  Fig.  428.  If  a  square-edged  tool  is 
used  it  creases  the  inner  plate,  and  if  this  should  prove  to  be  of  brittle  steel  it  might 
cause  a  failure  along  this  line.  On  all  plates  over  \  in.  thick  the  vertical  joints 
should  be  made  by  double  butt  straps,  as  shown  in  Fig,  427.  If  single  straps  are 
used  the  joint  is  no  stronger  than  a  lap  joint,  which  it  really  is.  When  these  are 
not  used,  one  of  the  plates,  where  three  plates  meet,  must  be  scarfed  down  to  a 
feather  edge,  and  this  should  be  done  by  heating  the  corner  of  the  plate  before 
putting  under  the  hammer.  This  heating  and  cooling  one  corner  of  a  steel  plate 
introduces  unknown  internal  stresses  into  the  plate,  especially  if  it  is  of  a  high  grade 
of  steel,  which  can  only  be  removed  by  annealing  the  plate.  The  butt  straps  avoid 
this  source  of  weakness  also. 

Bottom. — The  bottom  is  attached  to  the  sides  by  a  heavy  interior  angle-iron,  as 
shown  in  Fig.  425.  In  the  case  of  high  stand-pipes,  or  those  over  120  feet  high, 
there  should  be  two  of  these  angles  to  better  distribute  the  dead  weight  of  the  sides  on  the 

masonry,  as  shown  in  Fig.  429.  The  bottom  is  usually 
made  of  f-in.  plates,  and  this  is  attached  to  the  first 
ring  of  the  sides  and  thoroughly  calked,  and  then 
lowered  to  place  on  the  foundation.  This  should  have 
been  carefully  levelled  up  by  using  an  engineer's  level, 
and  then  covered  over  with  about  an  inch  of  soft  Port- 
land-cement mortar.  The  supports  are  then  removed 
and  the  lower  ring,  with  the  bottom  attached,  is  care- 
fully lowered  upon  its  cement  bed.    For  the  larger 


Fig.  429. 

sizes  the  bottom  must  be  held  up  at  one  or  more  points  on  the  interior  to  prevent  its  sagging 


Fig.  428. 


staN'b-pipes  and  elevated  tanks. 


433 


too  much.  This  is  done  by  attaching  permanent  eyes  to  the  bottom,  by  riveting  and  fasten- 
ing these  to  timbers  across  the  top  edge  of  the  vertical  sides.  If  the  bottom  were  allowed  to 
sag  it  would  destroy  the  calking. 

To  the  bottom  is  usually  attached  the  inlet  and  outlet  pipe,  which  is  commonly  one  and 
the  same ;  but  the  details  of  this  will  not  be  discussed  here. 

Man-holes  are  often  placed  in  the  lower  side  ring,  but  this  is  a  source  of  weakness.  It  is 
better  to  arrange  a  blow-out  pipe,  with  a  gate-valve  upon  it,  with  many  moutlis  uniformly  dis- 
tributed over  the  bottom  on  the  inside,  all  opening  downward,  and  properly  connected  with 
the  blow-out  pipe.  This  should  be  not  less  than  ten 
inches  in  diameter  for  a  stand-pipe  fifteen  feet  or  | 
over  in  diameter,  and  the  bottom  is  cleaned  by  | 
simply  opening  the  valve.  The  rush  of  water  out 
of  these  various  mouths  cleanses  the  entire  bottom. 
Even  though  perfectly  pure  ground-water  is  pumped 
into  the  stand-pipe,  a  considerable  amount  of  sedi- 
ment will  collect  at  the  bottom,  which  should  be 
cleaned  out  occasionally.  By  the  means  here  de 
scribed  it  is  not  necessary  to  empty  the  stand-pipe 
to  clean  it. 

Top  Angle. — To  prevent  the  top  from  collapsing 
from  the  force  of  the  wind,  a  strong  angle  iron,  not 
less  than  4"  X  4",  should  be  riveted  to  the  top, 
either  inside  or  outside. 

It  is  not  wise  to  cover  a  stand-pipe  where  the 
winters  are  severe.  A  thick  coating  of  ice  forms 
upon  the  sides,  which  may  become  loosened  as  warm 
weather  approaches,  and  if  this  is  done  suddenly,  its 
buoyancy  throws  it  with  great  force  upwards.  If  at 
such  a  time  the  stand-pipe  happens  to  be  ncarl}'  full 
of  water,  this  ice  column  may  be  forced  many  feet 
above  the  top.  If  a  roof  were  provided,  it  W(iuld 
probably  be  torn  off  by  such  an  action. 

463.  Ornamentation. — There  is  probably  no 
homelier  engineering  device  than  a  plain  steel  stand- 
pipe.  They  are,  however,  sometimes  made  more 
offensive  by  ill-advised  attempts  at  ornamentation 
There  seems  to  be  but  one  class  of  top  ornaments 
to  a  stand-pipe  which  are  appropriate.  This  is  a  com- 
bination of  a  cornice  and  a  sort  of  top  fencing  or 
open  fret-work.  The  cornice  is  not  full-surfaced,  but 
composed  of  plate  disks,  placed  near  together,  which 
in  profile  give  the  desired  curves.  Fig.  430  is  a  view 
of  the  stand-pipe  at  Jefferson  City,  Mo.,*  which  has 
such  an  ornamentation  as  here  described.  The 
photographic  view  here  shown  was  taken  from  a 
point  too  near  to  show  the  top  to  the  best  advan- 
tage. It  is  considered  an  ornament  to  the  city  by 
the  citizens,  and  yet  it  cost  but  a  few  hundred 

Fig.  430. 


*  Designed  by  Prof.  Johnson,  and  erected  in  i88S.  It  stands  on  a  prominent  liill  opposite  the  Capitol,  and  is 
visible  to  its  base  from  most  parts  of  the  city.  Its  dimensions  are  20  ft.  X  125  ft.  The  lop  ornamental  work  is  very 
inadequately  shown  in  this  cut  which  is  a  half-tone  from  a  retouched  photograph. 


434 


MODERN  FRAMED  STRUCTURES. 


dollars  to  transform  this  from  an  eyesore  into  a  pleasing  monument.  Since  these  struct- 
ures must  of  necessity  constantly  meet  the  public  gaze,  it  is  little  short  of  a  crime  to  make 
them  standing  embodiments  of  innate  ugliness. 

Elevated  Tanks. 

464.  General  Design. — Elevated  tanks  should  always  be  made  of  steel  plates,  and 
should  rest  on  a  wrought-iron  or  steel  trestle-tower.  Wooden  tanks  are  not  suflficientl}' 
permanent,  and  masonry  towers  over  forty  feet  high  are  unreliable.  It  is  practically  impossible 
to  make  a  tall  masonry  (brick  or  stone)  tower  act  as  one  monolithic  mass.  Its  strength  is 
always  problematic,  and  depends  so  largely  upon  the  character  of  the  workmanship  that  the 
designer  can  never  feel  assured  of  its  safety  under  the  heavy  loads  which  a  large  water  storage- 
tank  puts  upon  it. 

When  we  use  a  steel  tank  upon  a  trestle-tower,  it  is  difificult  to  support  a  flat  bottom  and 
to  distribute  this  load  horizontally  to  the  posts.  It  is  much  c4ieaper  and  more  scientific  to 
make  this  bottom  either  curved  or  conical.*  It  is  more  scientific  to  make  them  curved,  and  the 
most  practical  curve  is  the  circle.  The  bottom  should  therefore  be  spherical,  the  segment 
used  being  somewhat  less  than  a  hemisphere.  Such  a  bottom  can  be  readily  shaped  by  dishing 
the  plates  to  a  uniform  curve  under  a  steam-hammer.  This  increases  the  cost  somewhat,  but 
such  a  bottom  is  much  cheaper  than  a  flat  bottom  with  its  accompanying  floor  system. 

465.  Design  of  the  Tank. —  The  iJiickness  of  plates  to  use  in  the  sides  is  given  in  Fig.  423. 
The  thickness  of  the  plates  in  a  hemispherical  bottom  need  be  but  little  more  than  one  half 
that  of  the  bottom  ring  on  the  side.    The  computed  stress  on  the  bottom  plate  is  only  one 

pr 

half  that  in  the  sides  of  a  cylinder  subjected  to  the  same  internal  stress,  or  it  is  —  per  linear 

inch,  where  p  is  the  internal  pressure  per  square  inch,  and  r  is  the  radius  of  the  bottom  in  inches. 
If  a  segmental  bottom  is  used  less  than  a  full  hemisphere,  then  r  is  larger  here  than  for  the 
shell,  and  the  bottom  plates  must  be  correspondingly  heavier.  Allowing  the  same  unit  stress 
as  was  used  for  the  sides,  we  have,  ybr  thickness  of  bottom  plates, 

t  =  o.oooskr,  (8) 

where  /  =  thickness  of  bottom  plates  in  inches ; 
A  =  head  of  water  in  feet ; 
r  —  radius  of  bottom  in  feet. 
In  any  case  the  bottom  should  not  be  less  than      in.  thick. 

Riveting. — The  side  riveting  is  subject  to  the  same  rules  as  given  for  stand-pipes.  The 
bottom  sheets  should  all  be  single  riveted,  since  the  stress  on  these  joints  is  only  about  one 
half  what  it  is  in  the  side  plates.  The  attachment  of  the  bottom  to  the  sides  may  be  by 
a  single  or  double  row  of  rivets  as  preferred.  The  only  distortion  of  the  bottom  which  can 
occur,  due  to  a  fluctuation  of  height  of  water  in  the  tank,  is  a  vertical  deflection  of  the  whole 
bottom.  The  tendency  for  the  diametral  curve  to  change  its  shape,  on  account  of  its  not 
being  at  all  times  the  curve  of  equilibrium  for  that  particular  pressure,  is  fully  resisted  by  the 
circumferential  stress  in  the  bottom  plates.  Even  though  the  bottom  were  made  conical,  these 
plates  would  not  become  curved  on  a  diametral  section,  the  circumferential  stress  holding  them 
to  true  conical  surfaces. 

Concentration  of  tlie  Load  7ipon  the  Posts. — Fig.  433  illustrates  the  method  of  concentrating 
the  load  upon  the  columns.  The  bottom  is  here  shown  as  segmental,  and  because  this  pro- 
duces an  inward  pull  upon  the  shell  at  its  attachment,  two  strong  angle  irons  are  placed  here 
to  resist  this  compressive  stress. 

*  In  a  thesis  study  by  J.  M.  Raikes  Univ.  of  Mich.,  1896,  the  relative  weights  of  steel  in  a  tank  40  ft.  diam.  by 
40  ft.  high  were  for  flat  bottom,  with  beams  and  girders.  120,000  lbs.;  conical  bottom  and  circular  girder,  94,200  lbs.; 
spherical  bottom  and  circular  girder,  77,100  lbs.    {The  Technic,  1896.) 


STAND-PIPES  AND  ELEVATED  TANKS. 


435 


Let  P  =  the  compression  produced  in  the  shell  at  this  circle  by  this  action,  in  pounds; 
d  =  diameter  of  tank,  in  inches  ; 

W  =  total  weight  of  the  water  in  the  tank  plus  the  weight  of  the  bottom  itself  ; 
t  =  angle  of  bottom  with  the  vertical  at  the  attachment  circle. 

Then  we  have 


P=  ^  -  tan  t  •  —  =  OA^gW tan  t. 


(9) 


This  is  the  total  compression  in  the  two  angles  at  ^.   The  lower  sheet  of  the  shell  is  continu- 


FiG.  431. 


Pig.  432. 


436 


MODERN  FRAMED  STRUCTURES. 


ous  through  the  attachment  joint  at  A  to  the  bearings  on  the  posts  at  B.  Between  A  and  B  this 
sheet  is  stiffened  by  four  angles,  as  shown  in  elevation  and  plan  in  Fig.  433.  Between  these 
angles  and  the  top  of  the  post  a  plate  is  inserted  which  reaches  over  to  the  curved  bottom 
and  is  there  attached  as  shown.  The  top  of  the  column  is  covered  by  another  plate  which  is 
in  turn  supported  by  the  ends  of  the  column  members  and  angle  irons  in  some  suitable 
manner,  so  as  to  distribute  the  load  equally  over  the  members  composing  the  column. 

The  Roof  .—These,  tanks  are  not  so  high  as  to  cause  the  ice  to  interfere  with  a  roof,  and 
hence  they  should  be  covered.    The  roof  can  also  be  made  to  add  greatly  to  the  appearance 


Fig.  433. 

0/  the  tank,  as  shown  by  Figs.  431  and  432.*  A  curved  pagoda  roof  will  always  make  a  better 
appearance  than  a  conical  or  pyramidal  form. 


*  Fig.  431  is  from  a  design  prepared  by  Johnson  and  Flad,  St.  Louis,  in  1890,  for  the  western  suburbs  of  that  city. 
The  tank  shown  in  Fig.  432  was  designed  by  Mr.  Edw.  Flad,  C.E.,  M.  Am.  Soc.  C.E.,  for  Laredo,  Tex. 


STAND-PIPES  AND  ELEVATED  TANKS. 


437 


DETAIL  OF  CONNECTIONS 
HOR 

STRUTS  *  Tie  RODS. 


Relative  Dimensions. — The  more  nearly  the  tank  as  a  whole  approaches  the  spherical 
form  the  cheaper  it  will  be  in  proportion  to  its  volume.  It  makes  a  better  appearance,  how- 
ever, if  its  height  is  about  twice  its  diameter,  as  shown  in  Fig.  432.  Here  again  the  appearance 
should  be  a  prominent  and  determining  factor  in  preparing  the  design. 

466.  The  Trestle  Tower. — For  economy  the  trestle  legs  should  be  few  in  number.  A 
heavy  post,  or  column,  is  relatively  more  economical  than  a  light  one  of  the  same  length, 

because  —  is  less  for  the  large  post.    The  material  also  should  be  medium  steel,  or  steel  of 

from  62,000  to  70,000  lbs.  tensile  strength.  This 
material  is  now  so  cheap  that  there  is  no  longer 
any  object  in  using  cast-iron  in  columns  for  any  pur- 
pose. The  forms  may  be  either  Z  bars  or  channels, 
whichever  is  found  to  be  best  adapted  to  the  par- 
ticular details  used.  Probably  two  channels,  turned 
with  the  plane  sides  out,  and  latticed  on  two  sides, 
with  tie-plates  at  the  joints,  as  shown  in  Figs.  433  and 
434,  will  be  found  to  serve  every  purpose. 

The  smallest  number  of  posts  which  is  practi- 
cable is  four.  But  four  points  of  support  under  the 
tank  are  not  sufficient.  Mr.  Edw.  Flad,  C.E.,  has 
probably  made  the  best  solution  of  this  problem,  and 
it  will  be  here  described.  He  uses  four  posts,  as 
shown  in  Fig.  432,  and  from  each  post  extend  two 
braces  at  top,  thus  giving  twelve  points  of  support 
under  the  tank.  This  is  an  excellent  solution,  and 
causes  the  structure  to  present  a  very  satisfactory 
appearance,  but  it  leads  to  somewhat  complicated 
details  where  these  brackets  meet  the  posts  and  the 
tank.  The  details  of  these  connections  are  shown  in 
Fig.  433.  The  details  of  the  connections  for  struts 
and  ties  are  shown  in  Fig.  434.  In  this  figure  are 
also  shown  the  details  for  the  bases  of  the  posts,  the 
anchorages,  and  the  ladder,  which  is  attached  to  the 
outer  side  of  one  of  the  posts.  Tliese  details  were 
for  a  tank  40  feet  high  and  20  feet  in  diameter, 
having  a  capacity  of  85,000  gallons,  and  set  on  a 
trestle  80  feet  high. 

The  roof  has  a  wooden  framing,  set  on  iron 
brackets,  as  shown  in  Fig.  432.    An  air  space  24 
inches  high  was  left  between  the  top  of  the  tank  and  ^ 
the  sheathing  boards,  which  enables  the  tank  to  be  [■ 
entered. 

The  inlet  pipe  is  usually  provided  with  a  stuffing-  ~ 
box  to  prevent  any  excessive  stress  coming  on  the  ^"1°-  434- 

pipe  itself. 

467.  Relative  Cost  of  Stand-pipes  and  Elevator  Tanks.— When  the  great  saving  in 
the  foundation  is  taken  into  account,  the  elevated  steel  tank  will,  in  nearly  all  cases,  cost  less 
than  a  stand-pipe  of  the  same  efficiency,  and  it  is  usually  more  ornamental,  or  at  least  less 
offensive.  The  foundation  of  a  stand-pipe  will  cost  about  four  times  that  of  an  elevated  tank 
of  equal  capacity  for  service.    If  we  may  say  that  only  the  water  above  the  height  of  75  feet 


ast  Iron  BlaM 


438 


MODERN  FRAMED  STRUCTURES. 


from  the  ground  is  valuable  to  the  city,  then  an  elevated  tank  should  always  be  built  in  place 
of  a  stand-pipe.  When  large  quantities  of  water  are  to  be  stored  above  the  height  of  fifty 
feet,  then  the  elevated  tank  is  more  economical  for  equal  quantities,  and  the  greater  the 
height  at  which  the  storage  is  required  the  greater  is  the  economy  in  the  tank  design.  The 
great  objection  to  the  use  of  tanks  has  been  in  the  flat  floors  with  which  they  have  usually 
been  provided.  That  objection  is  now  removed  by  the  designs  for  curved  bottoms  here  pre 
sented.  Even  conical  bottoms  may  be  used,*  but  they  are  not  so  scientific  or  so  pleasing  in 
appearance  as  the  spherical  forms. 

An  elevated  tank  should  be  carefully  designed  by  a  competent  engineer,  and  working 
drawings  prepared.  When  so  designed  its  safety  is  more  secure  than  is  possible  in  the  case 
of  a  stand-pipe.  A  large  proportion  of  the  stand-pipes  in  this  country  have  been  designed  by 
contractors  and  sales  agents,  and  the  legitimate  results  of  such  designing  are  being  reached  in 
the  many  failures  of  such  structures  which  are  now  occurring. 

468.  Specification  for  Plate  Material  for  Stand-pipes  and  Elevated  Tanks. — The  fol- 
lowing clause,  or  its  equivalent,  should  be  inserted  in  all  stand-pipe  or  elevated-tank  specifi- 
cations : 

The  material  composing  the  plates  for  the  sides  and  bottom  shall  be  soft  steel  (preferably  open-hearth) 
having  a  tensile  strength  between  55,000  and  64,000  lbs.  per  square  inch  ;  an  elongation  in  eight  inches  of 
not  less  than  twenty  per  cent,  and  a  reduction  of  area  at  the  broken  section  of  not  less  than  forty  per  cent. 
Specimens  of  any  plate  having  a  widih  not  less  than  four  times  the  thickness,  when  heated  to  a  cherry  red 
and  quenched  in  water,  shall  bend  cold  through  180'  about  a  diameter  equal  to  its  thickness,  without  show- 
ing any  signs  of  failure  whatever. 

This  material  is  known  amongst  the  dealers  as  "  shell  steel,"  and  since  it  costs  perhaps 
a  quarter  of  a  cent  a  pound  more  than  "  tank  steel,"  which  is  made  by  the  Bessemer  process 
and  is  liable  to  be  very  hard  and  brittle,  the  contractor  is  pretty  sure  to  use  the  cheaper 
grade  if  special  care  is  not  taken  to  inspect  the  material  and  prove  its  character  by  actual 
tests. 


See  a  design  by  Mr.  Freeman  C.  Coffin,  C.E.,  Engineering  News,  Mar.  16,  1893. 


IRON  AND  STEEL  TALL  BUILDING  CONSTRUCTION. 


439 


CHAPTER  X}:VIII. 
IRON  AND  STEEL  TALL  BUILDING  CONSTRUCTION. 

469.  Modern  Building  Construction. — The  use  of  metal  in  the  construction  of  large 
buildings  has  increased  very  rapidly  in  the  last  few  years,  until  now  the  demand  for  structural 
steel  for  this  purpose  is  a  very  considerable  fraction  of  the  whole  output.  A  discussion  of  steel 
buildings  or  of  steel  construction  in  buildings  becomes  therefore  increasingly  important.  It 
perhaps  need  hardly  be  said  that  an  intelligent  use  of  iron  and  steel  in  this  field  requires  also 
a  considerable  knowledge  of  architecture,  and  more  especially  of  the  character  of  other  building 
materials  and  of  methods  of  using  them.  In  general  the  same  principles  apply  in  this  use  of 
metal  that  apply  in  all  other  framed  structures,  but  certain  problems  present  themselves  with 
much  greater  frequency  and  in  vastly  greater  variety  in  building  construction.  It  is  the  pur- 
pose of  this  chapter  to  discuss  some  of  these,  giving  a  few  illustrations  and  suggestions  that 
may  assist  the  inexperienced  to  do  intelligent  work  in  this  department. 

The  use  of  steel  to  so  great  an  extent  in  large  buildings  is  producing  a  new  type  of  con- 
struction, which  has  been  very  aptly  termed  the  "  sttel  skeleton  type  of  high  buildings."  The 
structural  steel  is  used  to  make  the  frame  of  the  building,  and  this  frame  should  be  strong 
enough  to  carry  the  loads  and  provide  rigidity  and  lateral  strength  to  the  structure.  The  con- 
struction is  typical  in  just  the  proportion  that  this  is  done  and  that  the  steel  frame  is  relied  on 
for.strength.  Several  buildings  have  been  constructed  which  are  of  this  pure  type.  They  have 
no  supporting  walls  about  them  anywhere ;  the  outside  covering,  of  whatever  material  it  may 
be— brick  or  terra  cotta  or  stone — and  all  the  interior  partition  walls,  are  carried  on  beams 
at  each  floor  level  and  are  self-supporting  only  one  story  in  height ;  while  all  the  weight  of 
the  building  is  carried  on  columns,  arranged  entirely  apart  from  each  other,  but  conveniently 
for  the  purpose. 

Between  this  pure  type  of  structure  and  the  old  construction,  in  which  all  the  strength  of 
a  building  was  in  its  heavy  masonry  walls,  there  is  every  degree  of  the  mixture  of  the  two. 
Even  some  of  the  sixteen-story  buildings  of  latest  date  have  solid  exterior  walls  of  masonry, 
while  the  inside  is  steel  construction.  Oftentimes  the  necessities  of  adjoining  buildings  or 
the  prejudices  of  owners  require  solid  masonry  party  walls  in  what  would  be  otherwise  a  steel 
structure  of  the  purest  type  ;  but  the  same  considerations  govern  the  designing  of  the  steel 
work  in  either  case,  and  the  use  of  the  supportini^  walls  only  increases  the  variety  of  the 
problems  to  be  solved.  This  characteristic  feature  of  steel  construction,  that  all  loads,  exterior 
and  interior  walls,  floors,  and  partitions,  are  carried  at  each  floor  level  by  the  frame  and  so 
taken  into  the  columns,  makes  it  necessary  to  calculate  the  weight  of  every  part  of  the  building 
and  determine  the  proper  dirstribution  of  all  the  loads  among  the  columns.  The  columns  are 
the  most  important  element  in  the  problem,  and  their  sizes  and  sections  must  depend  on  these 
results.  The  foundations  of  the  building  may  also  depend  on  these  loads,  and  especially  so 
if  the  structure  is  to  be  carried  on  a  yielding  soil  where  the  area  of  the  footings  must  be 
proportioned  to  them.  If  the  foundations  are  on  solid  rock  or  something  equally  as  good, 
it  may  be  sufficient  that  the  columns  simply  reach  their  resting-place.  The  exact  load  on 
each  bottom  column  is  more  important  where  piles  are  used  to  carry  them,  but  it  is  most  im 
portant  of  all  when  recourse  must  be  had  to  a  broad  bearing  on  a  compressible  soil. 


440 


MODERN  FRAMED  STRUCTURES. 


470.  The  Work  of  Designing  should  proceed  somewhat  as  follows:  (l)  Arrangement 
of  columns;  (2)  Arrangement  of  beams;  (3J Calculation  of  loads  on  columns;  (4)  Dimensions 
of  foundations ;  (5)  Design  of  spandrel  sections ;  (6)  Calculation  of  the  sizes  of  all  floor-beams 
not  already  fixed ;  (7)  Dimensions  of  the  columns ;  (8)  Wind  bracing ;  (9)  General  details  of 
connections. 

This  amount  of  work,  with  the  drawings  to  represent  it  and  specifications  to  cover  it,  are 
generally  done  at  the  expense  of  the  architect,  in  his  own  office,  or  by  some  consulting 
engineer.  Each  piece  of  iron  in  the  structure  should  finally  be  drawn  in  complete  detail, 
making  what  is  known  as  a  "shop  drawing,"  and  this  work  is  generally  done  at  the  expense  of 
the  contractor  subject  to  the  approval  of  the  architect  or  of  the  consulting  engineer. 

471.  Arrangement  of  Columns. — The  arrangement  of  the  columns  in  a  building  depends 
first  of  all  upon  the  plan  and  character  of  the  structure.  In  most  cases  there  will  be  certain 
points  at  which  columns  must  be  placed  on  account  of  the  shape  of  the  building,  the  interior 
arrangement  of  its  rooms,  its  staircases,  or  its  elevators,  and  regardless  of  constructional 
advantages  or  disadvantages.  In  planning  the  architectural  features  of  a  building,  however, 
the  construction  ought  to  be  kept  constantly  in  mind  and  not  made  too  subordinate,  for  the 
cost  may  be  kept  down  and  the  strength  and  general  good  character  of  the  framework  may 
sometimes  be  greatly  increased  by  very  slight  changes  in  construction,  to  permit  which 
architectural  features  can  be  varied  without  detriment  to  the  structure.  A  general  scheme  for 
the  arrangement  of  the  beams  in  the  floors  must  also  be  borne  in  mind  when  the  positions  of 
the  columns  are  fixed.  The  scheme  should  always  be  such  that  the  floor  loads  will  be  taken 
as  directly  to  the  columns  as  possible.  Very  few  rules  will  apply  to  all  cases.  Everything 
must  vary  with  the  necessities  of  the  case.  Even  economy  is  often  sacrificed  to  other  con- 
siderations. If  the  building  is  a  very  high  one,  the  bracing  of  the  structure  should  also  be 
considered  at  this  time,  for  some  of  the  columns  must  become  a  part  of  whatever  system  may 
be  used.  There  is  perhaps  no  part  of  the  work  of  designing  a  building  that  is  generally  as 
little  studied  and  yet  on  which  so  much  depends  regarding  economy  and  strength  and  the 
general  character  of  the  work,  as  this  one  in  particular,  and  none  in  which  a  large  experience  and 
a  trained  judgment  count  for  more.  The  more  completely  the  whole  problem  can  be  sized 
up  in  one  full  consideration  of  it,  the  more  satisfactorily  can  the  designer  arrange  his  columns 
and  at  the  same  time  plan  in  a  general  way  the  entire  framework  of  the  structure. 

472.  Arrangement  of  Beams. — As  soon  as  the  position  of  the  columns  is  determined, 
the  position  of  all  the  beams  in  the  floors  should  be  fixed  as  exactly  as  possible.  To  do  this 
it  will  be  necessary  to  calculate  the  size  of  the  principal  ones — that  is,  those  doing  the  most 
important  service. 

In  building  lore  the  word  "  girder"  has  come  to  mean  a  beam,  either  solid,  rolled,  or  built 
up  of  plates  and  angles  riveted  together,  which  is  used  to  support  joists,  and  the  word  "  beam  " 
is  used  to  signify  a  joist.  A  "  riveted  girder "  means  a  girder  made  of  plates  and  angles. 
A  "  girder  beam"  means  a  girder  made  of  a  solid  rolled  beam.  A  "  spandrel  beam  "  is  a 
common  term  for  a  beam  carrying  a  portion  of  the  exterior  wall  of  a  building.  A  "double 
girder  "  signifies  the  use  of  two  rolled  beams  in  a  girder.  These  expressions  will  be  used  with 
these  meanings. 

It  is  not  necessary  at  this  time  to  calculate  the  size  of  the  smaller  or  unimportant  beams, 
because  the  exact  arrangement  of  the  beams  is  needed  at  this  juncture  only  to  determine  the 
true  distribution  of  the  floor  load  on  the  columns.  The  fewer  connections  between  a  load  and 
the  column  which  carries  it,  the  better  the  construction  ;  therefore  the  girders  should  always 
connect  the  columns,  if  possible,  and  multiplicity  of  details  in  arrangement  should  be  avoided. 
The  arrangement  of  elevators  and  stairways  and  pipe  spaces  in  large  office  buildings,  and 
sometimes  of  the  machinery,  makes  it  impossible  to  get  the  ideal  plan;  but  a  studied  effort 
in  the  right  direction  will  often  greatly  simplify  what  at  first  seemed  a  necessarily  complicated 


IRON  AND  STEEL  TALL  BUILDING  CONSTRUCTION. 


441 


design.  The  arrangement  of  the  beams  may  be  varied  to  some  extent  with  the  method 
of  fire-proofing  to  be  employed ;  so  it  is  important  to  have  a  plan  for  the  completion  of  the 
floor  also  in  mind  at  this  time.  There  are  a  good  many  schemes  for  fire-proof  floors,  but  not 
very  many  in  actual  service.  In  many  large  buildings  the  floor  framing  is  made  regardless  of 
the  arrangement  of  partitions,  in  order  that  these  may  be  arranged  and  rearranged  from  time 
to  time  to  suit  the  taste  of  those  who  use  the  rooms,  but  in  some  buildings  built  for  speci:il 
uses  few  partitions  are  required,  and  in  such  cases,  or  where  partitions  are  to  be  particularly 
heavy,  it  is  best  to  arrange  the  framing  so  that  a  beam  or  girder  will  come  directly  under  the 
load. 

In  most  large  buildings  the  framing  of  many  of  the  floors  will  be  the  same.  A  drawing 
should  be  made  of  this  typical  floor,  and  also  drawings  of  those  that  differ  from  it  ;  they  should 
be  made  as  simple  as  possible.  A  small  square  or  a  circle  is  enough  to  show  the  position  of 
the  columns,  and  a  single  line  is  enough  to  show  the  position  of  the  beams.  Architectural 
draughtsmen  depend  much  on  reference  to  scale  to  get  sizes  and  dimensions  from 
their  drawings.  All  drawings  of  metal  construction  should  preferably  be  made  to  a 
scale ;  dependence  should  be  placed  however,  entirely  on  figured  dimensions,  and  all 
figures  should  be  exact.  There  can  be  no  approximations  in  iron-work.  The  distance 
between  the  centres  of  two  beams  is  not  about  5  feet,  it  is  exactly  5  feet  2|  inches,  and  this 
exactness  should  be  exercised  on  all  iron  draivings.  These  j)lans  should  show  the  building 
lines,  the  centre  lines  of  all  columns  and  their  relations  to  the  centre  lines  of  all  beams  and 
girders.  When  this  is  done  and  the  general  construction  of  the  building  is  determined,  then 
can  begin  the  actual  work  of  calculating  the  loads  on  the  columns. 

473.  Economy  of  Wide  Spacing  of  Floor-beams. — In  carrying  the  floor  loads  to 
the  columns,  which  must  always  be  done  by  the  action  of  beams  in  some  form,  it  is  an  evident 
economy  to  allow  the  floor  material,  whether  composed  of  tile,  concrete,  or  plank,  to  carry  the 
distributed  load  as  far  as  it  can  do  so  with  safety.  A  stronger  floor  system  can  therefore  be 
used  with  wider  spacing  between  floor-beams.  Since  the  strength  of  these  beams  increases  in 
a  general  way  with  the  square  of  their  depth,  it  would  be  economical  to  place  them  farther 
apart,  thus  increasing  the  load  on  each  beam,  and  using  a  greater  depth.  Since  these  beams 
are  carried  at  their  ends  by  girders  resting  on  the  columns,  it  is  desirable  also  to  so  arrange 
them  that  the  bending  moment  on  these  girders  will  be  the  least.  This  requires  that  there 
be  an  odd  number  of  panels,  or  spaces  between  beams,  on  each  girder.  This  brings  two 
beams  symmetrically  on  the  middle  portion,  between  which  the  bending  moment  is  constant. 

Example. — Take  a  floor  space  20  x  24  feet,  to  be  carried  on  columns  at  the  corners.  Assume  two 
arrangements  of  beams:  First,  seven  beams  (six  panels)  20  feet  long,  resting  on  a  plate  girder  24  feet  long. 
Second,  four  beams  (three  panels),  supported  in  the  same  manner.  In  the  one  case  the  beams  are  4  feet 
apart,  and  in  the  other  case  they  are  8  feet  apart.*  Assume  a  total  unit  loading,  dead  and  live,  of  180  lbs. 
per  square  foot. 

First  Case. 

The  bending  moment  on  the  beams  (counting  one  outside  beam,  and  one  of  the  girders,  each  fully 

loaded  as  though  other  similar   areas  of  floor  space  surrounded    this  one  on   all   sides)  would  be 

4  X  180  X  20  X  20  /  r 

 =  36,000  ft.-lbs.     This  would  take  a  ten-inch  beam  weighing  27.75  'ds.  per  toot  lor  a 

8 

fibre  stress  of  16,000  lbs.  per  square  inch. 

The  bending  moment  on  the  girder  would  be  259,200  ft.-lbs.  If  it  have  a  total  depth  of  30  inches,  with 
three  eighths  web  and  4  x  4  x  f "  angles,  the  fibre  stress  will  be  again  16,000  lbs.  per  square  inch,  one 
sixth  of  the  area  of  the  web  being  added  to  the  effective  area  of  the  angles  to  give  total  flange  area.  The 
weight  of  this  beam  would  be  78  lbs.  per  foot,  or  1872  lbs.  for  the  24-foot  girder.  Counting  six  beams  and 
one  girder  as  belonging  to  this  elementary  area,  we  have  a  total  weight  of  5200  lbs. 

If  two  rolled  beams  be  used  in  place  of  the  plate  girder,  it  would  require  15-inch  beams  weighing  72  lbs. 


*  A  concrete  arch  will  readily  span  8  feet. 


MODERN  FRAMED  STRUCTURES. 


per  foot,  or  3456  lbs.  for  the  two  beams  to  act  as  one  girder.  When  these  are  used  the  total  weight  of  iron 
to  carry  this  area  with  four-foot  spacing  of  beams  would  be  6790  lbs. 

Second  Case. 

Here  the  beams  are  placed  8  feet  apart.  The  bending  moment  in  each  beam  would  be  72,000  lbs.,  re- 
quiring a  15-inch  beam  weighing  41  lbs.  per  foot  for  a  fibre  stress  of  16,000  lbs.  per  square  inch. 

There  are  now  but  two  beams  upon  the  girders,  giving  a  bending  moment  of  230,400  ft. -lbs.,  wliic  li 
would  require  a  plate  girder  30  inches  deep,  f-inch  web,  and  4"  x  1^"  x  f"  angles,  the  whole  weighing  74 
lbs.  per  foot,  or  1776  lbs.  for  the  girder. 

If  two  I  beams  were  used,  as  before,  it  would  require  15-inch  beams  weighing  60.5  lbs.  per  foot,  or 
2904  lbs.  for  the  two  beams  composing  the  girder. 

In  this  case,  therefore,  we  have  three  beams  weighing  2460  lbs.,  which  with  the  plate  girder  make  a 
total  of  4236  lbs.,  or  with  the  double  girder  5364  lbs. 

Thus  when  the  30-inch  plate  girders  are  used,  the  weight  of  iron  for  the  4-foot  spacing  is  5200  lbs.,  as 
against  4236  lbs.  for  the  8-foot  spacing,  or  a  saving  of  18.5  per  cent  of  the  iron  used  in  the  floor  system  with 
4-foot  spacing. 

When  the  double  15-inch  girders  are  used,  we  have  a  total  weight  of  6790  lbs.  for  the  4-foot  spacing,  a.§ 
against  5364  lbs.  for  the  8-foot  spacing,  or  a  saving  of  21  per  cent  of  the  iron  in  the  floor  system  for  the 
4-foot  spacing.  These  are  the  percentages  of  saving  made  by  using  8-foot  instead  of  4-foot  spacing.  The 
weights  of  I  beams  have  been  taken  exactly  those  required  for  the  given  bending  moments  in  all  cases,  as  this 
is  necessary  to  obtain  a  fair  comparison  of  weights  from  a  single  example. 

If  the  flooring  for  the  8-foot  spacing  is  more  expensive  than  that  for  4  feet,  an  allowance  can  be  made 
for  this  and  the  final  net  saving,  due  to  the  wider  spacing,  computed. 

474.  Calculation  of  Column  Loads. — The  column  loads  should  be  divided  into  two 
classes,  dead  loads  and  live  loads.  The  actual  permanent  weight  of  the  structure  itself  makes 
the  dead  load  ;  and  the  estimated  weight  of  the  people  that  will  at  any  time  enter  the  build- 
ing, together  with  furniture,  movables,  stocks  of  goods,  etc.,  make  the  live  load.  If  machinery 
is  permanently  fixed,  it  should  be  counted  as  dead  load  ;  if  it  is  movable,  it  is  usually  rated  as 
live  load. 

Dead  loads  may  generally  be  divided  as  follows : 

Weight  of    (i)  Floors  ; 

(2)  Partitions; 

(3)  Vaults  ; 

(4)  Metal  Columns ; 

(5)  Column  Coverings; 

(6)  Exterior  Walls ; 

(7)  Windows ; 

(8)  Elevators ; 

(9)  Permanent  Machinery; 

(10)  Water  Tanks ; 

(11)  Plumbing  and  Heating  Fixtures. 

The  weight  of  floors  is  usually  reckoned  by  the  square  foot.  The  number  of  feet  sup- 
ported by  each  column  should  be  determined,  and  this  amount  multiplied  by  the  weight  per 
foot  gives  the  desired  figure.  The  accuracy  of  the  work  depends  very  greatly  upon  this 
distribution  of  supported  areas.  If  the  floor  plan  is  extremely  simple,  this  may  be  done  very 
easily  and  sometimes  by  simple  observation,  but  in  most  plans  there  is  somewhere  a  com- 
plication or  irregularity,  and  the  safest  way  is  to  figure  the  areas  directly  on  a  copy  of  the  floor 
plans.  There  are  several  ways  in  which  this  can  be  done.  The  whole  area  may  be  divided 
into  rectangles  cornering  at  the  columns.  Each  of  these  rectangles  can  then  be  subdivided 
and  the  number  of  square  feet  in  each  part  can  be  noted  on  the  drawing  next  the  cplumn  to 


IRON  AND  STEEL   TALL  BUILDING  CONSTRUCTION. 


443 


which  that  particular  part  is  tributary.  It  may  be  done  also  by  computing  the  area  carried  by 
each  individual  beam,  then  the  exact  reactions  of  both  the  beams  and  the  girders,  keeping  all 
the  results  in  square  feet.  Either  of  these  or  other  methods  may  require  considerable  figur- 
ing where  the  floor  plan  is  very  irregular,  but  the  results  can  be  made  always  definite  and 
accurate.    Supported  column  areas  should  never  be  estimated  or  guessed  at. 

The  weight  of  the  floor  per  square  foot  also  should  be  determined  as  closely  as  possible. 
Practice  differs  considerably  in  different  cities,  and  somewhat  among  different  architects  in  the 
same  city.  The  most  approved  floor  in  Chicago,  where  this  construction  has  perhaps  reached 
its  greatest  development,  is  shown  in  Fig.  435. 


Fig.  435. 


In  this  construction  the  floor  weight  is  made  up  of  the  following  items  :  Iron,  tile  arch, 
concrete  filling,  plastering,  and  wood  floor.  The  iron  consists  of  the  beams,  tie-rods,  girders, 
connections,  etc.,  which  may  be  averaged  per  square  foot.  In  ordinary  floors  it  will  be  from  8 
to  12  pounds.  It  may  be  easily  figured  from  the  floor  plans  so  that  it  will  not  vary  more  than 
a  fraction  of  a  pound.  The  depth  of  the  concrete  should  be  known,  also  the  weight  of  it.  A 
concrete  made  of  cinders  and  lime  and  a  small  amount  of  cement  is  one  of  the  lightest  and 
best  for  this  purpose.  Such  a  concrete  weighs  about  72  lbs.  per  cubic  foot.  Partitions  should 
be  of  some  entirely  fire-proof  material,  light  and  ea.sy  to  put  in  place.  Partitions  in  office 
buildings,  hotels,  etc.,  should  be  built  after  the  floor  is  laid  and  at  least  the  first  coat  of  plaster 
is  on.  Then  they  can  be  taken  down  and  rebuilt  in  other  places  without  leaving  any  per- 
manent marks  of  the  change.  Where  they  are  built  in  this  way,  it  is  best  to  calculate  the 
entire  weight  of  the  partitions  on  a  floor  and  find  the  average  per  square  foot.  In  such  cases 
this  average  can  be  included  in  the  floor  load.  In  most  buildings  it  will  not  be  greatly  in 
error.  If  the  partitions  are  sure  to  be  permanent  and  the  location  is  already  fixed,  a  separate 
distribution  of  their  weight  is  of  course  possible  and  preferable.  The  thickness  of  the  plaster 
may  be  kept  quite  uniform  by  making  it  true  to  the  woodwork  about  the  doors,  and  the 
weight  of  the  partition  stuff,  together  with  the  weight  of  the  plaster,  will  make  up  the  whole 
weight  of  the  partition.  Doors  can  be  calculated,  or  sample  ones  can  be  actually  weighed  and 
averaged.  The  common  vaults  in  office  buildings  are  only  very  small  rooms  with  iron  doors 
and  combination  locks.  In  such  cases  tlieir  weight  may  be  lumped  with  that  of  the  partitions. 
Bank  vaults  are  made  of  solid  brick  walls  .steel  lined,  and  they  must  have  special  treatment. 
The  weight  of  the  column  itself  will  have  to  be  estimated  to  some  extent,  but  experience  can 
make  a  very  accurate  estimate  possible.  The  column  covering  consists  of  the  fire-proofing, 
generally  some  kind  of  tile,  and  the  finish,  which  is  usually  plaster,  but  sometimes  marble  or 
other  material.  The  calculation  of  the  weight  of  the  exterior  walls  involves  additional 
problems  in  reactions.  The  "curtain  wall"  is  that  part  of  the  exterior  wall  extending  from 
the  line  of  the  window  cap  of  one  story  to  the  line  of  the  window  sill  of  the  next  story  above. 
This  part  of  the  wall  is  aji  evenly  distributed  load  on  the  spandrel  beam,  while  the  rest  of  it  is 
cut  up  into  separate  concentrated  loads  which  may  or  may  not  be  symmetrical  about  column 
centres.  Exterior  walls  are  made  of  brick,  terra  cotta,  stone,  tile,  metal,  or  combinations  of 
these  materials.  The  only  way  to  find  the  correct  weight,  of  course,  is  to  know  exactly  what 
is  to  be  used  and  ju.st  how,  so  that  cubic  contents  can  be  figured.  The  floor  construction 
sometimes  projects  into  the  curtain  walls.    In  such  cases  care  must  be  used  to  insure  that 


444 


MODERN  FRAMED  STRUCTURES. 


this  portion  of  floor  is  not  put  in  as  part  of  the  wall.  The  weight  of  windows  must  be  doubled 
to  allow  for  counter-weights.  All  the  other  dead  loads  are  single,  and  most  of  them  occur 
only  on  special  floors. 

The  live  loads  used  in  buildings  vary  from  almost  nothing  to  300  or  400  lbs.  per  square 
foot.  The  beams  should  be  calculated  to  carry  the  whole  of  the  unit  taken,  but  in  many  cases 
it  is  wise  to  reduce  the  rate  in  figuring  the  girders  and  the  loads  on  the  columns.  The  live  load 
taken  on  columns  in  ofifice  buildings  vary  from  20  to  50  lbs.  per  square  foot  of  floor  ;  sleeping- 
rooms  in  hotels  do  not  require  more  than  10  to  20  lbs.,  while  60  to  80  lbs.  per  square  foot 
ought  to  be  allowed  for  floors  where  people  may  assemble  in  crowds,  and  warehouses  some- 
times require  very  much  more.  This,  of  course,  is  in  addition  to  the  dead  loads.  These 
loads  must  be  figured  for  every  floor,  and  tabulated  so  that  the  accumulated  load  can  be  de- 
termined for  any  particular  column  in  any  particular  story.  When  the  work  is  completed, 
the  column  loads  are  not  only  at  hand,  but  the  loads  on  the  foundations  can  also  be  easily 
determined  from  the  schedule. 

475.  Dimensions  of  the  Foundations. — A  general  discussion  of  foundations  is  not  here 
intended,  but  steel  has  been  used  so  much  in  foundations  that  bear  on  a  yielding  strata,  that 
such  foundations  seem  almost  a  part  of  the  steel  construction  of  buildings.  Most  of  the  great 
buildings  in  Chicago  have  foundations  resting  on  soft  clay.  In  order  to  secure  an  equal  settle- 
ment in  those  buildings,  it  is  generally  advisable  to  omit  the  live  (temporary)  load  entirely, 
and  then  the  basement  column  load  minus  the  live  load  and  plus  the  weight  of  the  founda- 
tion itself  will  equal  the  permanent  load  on  the  clay. 

Figs.  436  and  437  show  a  plan  and  section  of  such  a  foundation.  The  area  covered  is 
a\ways  in  proportion,  or  should  be,  to  this  load  on  the  clay.  The  following  formula  may  be 
found  useful  in  calculating  the  beams. 

Let  y  =  the  width  of  the  projecting  area ; 
P=  total  load  on  the  top, 

=  total  reaction  at  the  bottom  ; 
a  =  the  length  of  distribution  of  the  load  at  the  top ; 
M=  maximum  bending  moment. 

The  use  of  a  cast  base  under  a  column  as  shown  in  Fig.  436  has  several  advantages  and 
is  generally  economical.    These  bases  are  made  strong  enough  to  carry  the  entire  load  on 
their  perimeters,  and  in  such  cases  the  maximum  bending  moment  in  the  beams  comes  at  the 
edge  of  the  base-plate.    The  length  of  the  beam  =  2^     a,  and  the  total  reaction  of  the  pro- 
Pv 

jecting  area  =  ^  ^  and  this  multiplied  by  one  half  the  width  of  the  projecting  area  is  the 
bending  moment  at  the  edge  of  the  plate  or 

M=^^^  (.) 

2(2/  -\-a)  ^  ' 


Either  M  ox  y  may  be  the  unknown  quantity. 

In  the  case  shown  in  Figs.  436  and  437,  in  the  top  course 


P=  1,233,270  lbs.; 
«  =  4'  6"  ; 


and  these  introduced  into  the  formula  make  M  =  1,452.760  ft.-lbs. 


IRON  AND  STEEL  TALL  BUILDING  CONSTRUCTION, 


Fig.  437. 


446 


MODERN  FRAMED  STRUCTURES. 


If  the  base-plate  is  only  strong  enough  to  uniformly  distribute  the  load  over  the  area  a, 
the  maximum  bending  moment  in  the  beams  is  at  the  centre,  and  then  if  /  =  the  length  of 
the  beam, 

M=^{l-a)  (2) 


It  will  be  seen  by  computation  and  comparison  that  this  condition  will  increase  the  weight 
of  metal  required  in  the  top  layer  in  the  example  given  about  35  per  cent. 

The  Moment  of  Resistance,  M^,  in  foot-pounds,  for  any  given  beam  may  be  obtained  as 
follows : 

Let  f=  ultimate  allowable  fibre  strain  per  sqaare  inch; 
/=  moment  of  inertia,  in  inch  units  ; 
y  =  one  half  the  depth  of  the  beam. 

Then  from  Eq.  (i),  Art.  121, 

M,  =  ^,    .    .    .  (2a) 


The  value  of  /  for  any  beam  is  always  given  by  the  manufacturer  of  it  in  some  published 
book  or  list.  The  fibre  stress,  f,  for  foundation  work  is  usually  taken  at  20,000  lbs.  per  square 
inch.  Steel  rails  are  sometimes  used  instead  of  beams,  but  are  not  often  stressed  as  high. 
Sometimes  these  foundations  are  more  than  two  layers  of  steel  deep.  Theoretically  the  sum 
of  the  moments  of  resistance  of  all  the  beams  in  one  direction  should  equal  the  bending 
moment  as  given  in  eq.  (i),  irrespective  of  the  number  of  layers.  In  practice,  however,  where 
there  are  more  than  two  layers,  so  much  steel  is  hardly  needed.  The  beams  are  always  bedded 
completely  in  Portland-cement  concrete,  and  the  friction  of  the  beams  on  the  cross-layers, 
together  with  the  adhesion  of  cement  and  iron,  tend  to  unite  the  whole  into  a  sort  of  com- 
pound beam,  the  total  moment  of  resistance  of  which  is  much  greater  than  the  sum  of  "the 
moments  of  the  separate  beams. 

Two  or  more  of  these  footings  are  often  combined  into  a  single  one  where  the  area  can 
only  be  obtained  in  that  way,  or  for  other  reasons.  Fig.  438  shows  an  elevation  sketch  of  such 
a  combined  footing.  The  centre  of  gravity  of  such  areas  should  coincide  with  the  centre  of 
gravity  of  the  loads. 


'<.     a — >  a- 
'      I  .     '  J.  :     .  I  I 


— >|<- 
p 

1 
1 

|< — m--=* 
1 

P 

P 

ij  T  T  T  f  I^"T  T  I  T  I  TT  M  r\  T  li 

\  -;  i;^;  »  p"=  p  +  p'   ^1 

Fig.  438. 

To  find  the  maximum  bending  moment  on  the  long  beams  we  are  obliged  to  compute 
three  moments  and  compare  them.  In  Chapter  VIII  it  was  shown  that  the  bending  moment 
is  a  maximum  where  the  shear  is  zero.  In  this  case  there  are  three  such  sections,  antl  it  will 
be  necessary  to  compute  the  moment  at  each  one  to  find  which  is  the  greatest.  The  moments 
under  the  columns  will  be  positive,  causing  convexity  downwards,  while  that  at  the  centre  is 
negative. 


IRON  AND  STEEL   TALL  BUILDING  CONSTRUCTION. 


447 


To  find  the  distance  from  the  left  end  to  the  centre  of  gravity  of  the  loads,  we  have,  Fig. 

438, 

^^^PV+i)  

If  the  area  is  rectangular,  this  gives  the  centre  of  it.  If  it  is  trapezoidal,  its  centre  of  gravity 
must  be  at  this  point. 

P  P'  P-\-  P' 

\[  p  —      p'  —  — ,  and  p"  —  — ~. — ,  as  in  Fig.  438,  then  to  find  the  distance  from  the  left 

end  of  the  bottom  beams  to  the  sections  where  the  shear  is  zero,  these  distances  being  called 
d^,d^,  and     respectively,  we  have 

d,p"  =  {d,-m)p,    or  ^.=j^;  (4) 

^./'  =  P,   or   ^.=^;  (5) 

d,p"  =  P+id,-{m  +  a  +  n)-\p',   or      ^      +  ^+ ^0  /  -      ....  (6) 

The  bending  moments  at  these  points  are  readily  found  by  taking  the  moments  of  the  external 
forces  on  one  side  of  the  point  about  that  point.  Thus  the  bending  moment  at  the  first  max- 
imum point  is  '■ 

M,=qi-M^,  (;> 

M.  =  p{d-^);  (8) 


2 


M,  =  ^'  -  /X^.  -  ^)  -f'^''  -  "•-  "  -   (9) 


In  general  will  be  small,  except  where  P  and  P'  are  near  the  ends  of  the  beams,  and 
the  maximum  moment  will  usually  be  either       or      ,  whichever  is  the  heavier  load. 

If  the  cast  bases  are  strong  enough  to  carry  the  loads  on  their  perimeters,  and  the  long 
beams  are  in  the  top  course,  the  values  of  M^  and  may  be  reduced  ;  Al^,  however,  would 
not  be  changed. 

The  problems  of  this  class  are  sometimes  exceedingly  complex,  as  when  foundations  can- 
not extend  beyond  the  lot  line,  or  when  a  sewer  or  other  obstruction  is  encountered,  and  the 
study  of  economy  in  designing  offers  a  wide  field  of  investigation  along  this  line.  Sometimes 
it  happens  that  the  clay  loads  are  so  great  and  the  limits  so  narrow  that  it  is  impossible  to  find 
area  enough  without  overloading  the  clay.  In  such  cases  cither  the  dead  load  of  the  building 
must  be  lightened  or  the  columns  must  be  rearranged.  In  either  case  much  of  the  work 
already  done  must  be  done  again. 

476.  Design  of  Spandrel  Sections. — The  term  "spandrel  section"  commonly  means  a  ver- 
tical cross-section  through  the  exterior  wall  of  a  building,  showing  the  construction  between  the 
top  of  one  window  arid  the  bottom  of  the  next  one  above  it.  In  most  cases  the  spandrel  beam 
must  carry  the  fiooi*  and  support  the  wall.  In  order  to  intelligently  design  the  iron  in 
a  spandrel  section,  it  is  quite  necessary  to  have  the  floor  plan  already  arranged,  and  to  know 
exactly  how  the  wall  is  to  be  built.  The  architectural  work  proper  must  be  practically  fin- 
ished. Cornice  lines,  reveals,  projections,  the  dimensions  of  terra  cotta  or  stone  trimmings, 
the  relative  heights  of  window  caps  and  floor  lines,  the  exact  position  of  the  wall  lines  in  the 


MODERN  FRAMED  STRUCTURES. 


IRON  AND  STEEL  TALL  BUILDING  CONSTRUCTION. 


449 


rooms  both  above  and  below,  and  in  fact  all  the  exact  data  pertaining  to  the  construction,  are 
needed  to  design  the  iron  in  the  best  possible  manner.  The  iron  must  not  only  not  be  exposed 
anywhere,  but  it  should  be  far  enough  from  the  lines  of  exposure  on  all  sides  to  be  thoroughly 
fire-proof.  No  part  of  a  building  is  more  exposed  to  a  great  heat  in  case  of  fire  than  this,  for 
window  openings  become  draft-holes,  through  which  the  flames  can  play  with  greatest  fury. 
The  iron  must  also  not  only  be  strong  enough  to  carry  the  load,  but  it  must  be  so  arranged 
that  both  the  wall  and  the  floor  shall  be  fully  supported. 

If  the  floor  arch  rests  directly  on  the  spandrel,  either  the  bottom  flange  of  the  beam  must 
be  on  the  same  level  as  the  floor-beams,  or  an  angle  must  be  riveted  to  the  web  of  the  beam, 
as  shown  in  Figs.  439  and  440.  When  the  spandrel  beam  serves  as  a  girder,  it  can  be  placed 
as  high  or  as  low  as  desired.   This  is  illustrated  in  Figs.  441  and  442.   Terra  cotta  is  used  to  a 


— 

I 

Fig.  441. 

very  great  extent  outside  as  a  finishing  material  in  this  type  of  construction.  It  is  used  for  the 
entire  outside  finish,  or  it  is  used  for  window  caps  and  sills  and  in  various  other  ways  in  con- 
nection with  pressed  brick.  The  proper  idea  is  to  get  the  iron  as  well  under  the  terra  cotta  as 
possible,  and  where  that  cannot  be  done,  to  arrange  the  iron  so  that  the  bottom  course  in  each 
spandrel  can  be  readily  suspended.  Fig.  439  shows  such  construction.  The  terra-cotta  blocks 
making  the  window  cap  are  drawn  into  position  close  against  the  iron-work  with  hook  bolts, 
so  that  each  has  its  own  support.  The  next  piece  is  anchored  with  slight  iron  rods  in  position 
.  with  one  edge  resting  on  the  horizontal  flange  of  an  angle,  so  that  none  of  its  weight  comes  on 
the  suspended  blocks.  Fig.  440  shows  another  construction  where  the  reveal  is  less  and  the 
window  cap  is  notched  so  as  to  ride  directly  on  the  flange  of  the  Z  bar.  Care  should  be 
exercised  in  all  cases  to  make  the  connections  in  some  way  that  will  prevent  these  angles  and 


45° 


MODERN  FRAMED  STRUCTURES. 


7j  bars,  or  any  other  iron  that  may  be  used  for  the  purpose,  from  deflecting  or  twisting  so  as  to 
crack  the  wall  after  all  the  work  is  done.  Such  cracking  may  not  endanger  the  construction 
in  any  way,  but  it  will  be  sure  to  mar  the  appearance  of  the  building,  and  it  can  be  entirely 
avoided.  Brick  can  be  used  without  terra  cotta  by  turning  a  flat  arch  for  a  window  cap  and 
taking  all  the  load  above  it  directly  on  the  iron  or,  indeed,  by  suspending  the  brick  directly 
over  the  window,  in  which  case  special  brick  are  required.  When  stone  window  caps  are 
used,  they  can  also  be  carried  directly  on  the  iron,  but  more  generally  the  window  cap  itself  is 
'made  self-supporting,  and  the  load  above  is  taken  on  the  iron,  as  shown  in  Fig.  441.  Where 
the  distance  between  columns  is  very  great  the  beams  should  not  be  strained  as  high  as  usual 
on  account  of  deflection,  and  for  the  same  reason  deep  beams  are  preferable  to  shallow  ones. 
Deflection  should  be  kept  at  a  minimum  in  all  spandrel  work. 


:  '3 

Fig.  442. 

In  calculating  the  loads,  that  part  of  the  wall  between  the  lower  line  of  the  window  cap 
and  the  top  line  of  the  window  sill  next  above  is  almost  always  evenly  distributed.  The 
muUions  between  the  windows  and  the  windows  themselves  sometimes  must  be  treated  as 
separate  loads,  while  the  floor  loads  coming  on  these  beams  may  come  under  either  class. 

477.  Calculation  of  Beams. — The  best  method  of  determining  the  size  of  beams  is  that  of 
moments.    To  use  this  method  readily  it  is  necessary  to  have  a  table  showing  the  values  of 
for  each  section  of  beam  which  there  is  any  possibility  of  using.    Such  a  table  can  easily 
be  made,  using  formula  (2)  for  the  calculations.    The  value  of  p  is  usually  taken  at  16,000  lbs. 

Determine  the  bending  moment  of  each  beam,  taking  all  distances  in  feet,  then  select  a 
beam  from  the  table  whose  moment  of  resistance  is  not  less  than  the  calculated  bending 
moment.  This  method  of  work  would  be  understood  anywhere,  and  is  applicable  to  all  cases 
that  can  possibly  occur. 

Dimensions  of  Columns. — In  actual  practice  the  treatment  of  columns  varies 
greatly.  This  is  mostly  due  to  the  following  circumstances  :  The  formulje  for  the  strength  of 
columns  do  not  agree.   The  underlying  principles  seem  to  be  sufficiently  established,  but  every 


IRON  AND  STEEL   TALL  BUILDING  CONSTRUCTION. 


authority  has  his  own  treatment  of  them,  his  own  form  of  expression  and  his  own  nomencla- 
ture. To  some  extent,  also,  they  are  empirical,  containing  factors  entirely  dependent  upon 
the  results  of  actual  tests,  and  these  have  given  rise  to  further  differences.  All  common 
formulae  are  based  on  a  condition  of  ideal  loading  which  cannot  always  be  obtained  in  build- 
ing construction ;  indeed,  it  would  be  nearer  the  truth  to  say  it  is  rarely  obtained.  There  is 
also  a  lack  of  full-sized  tests  right  along  the  line  of  these  irregularities  of  loading.  The  tests 
that  have  been  made  are  not  full  enough  to  properly  show  the  relative  value  of  the  different 
sections  in  use,  and  are  not  conveniently  available  to  the  profession  at  large.  All  this  helps 
to  explain  the  lack  of  uniformity  in  the  estimate  of  column  sections  and  methods  of  calculat- 
ing them. 

In  the  treatment  of  columns  given  in  Chapter  IX  three  kinds  of  compressive  stresses  are 
pointed  out  to  which  the  concave  side  of  the  bent  column  is  subjected  :  that  uniformly  dis- 
tributed over  the  section,  that  due  to  eccentric  loading,  and  that  due  to  the  flexure.  In  the 
derived  formulae  the  second  of  these  elements  is  omitted  because  there  can  be  no  eccentricity 
in  ideal  loading,  and  so  it  is  in  Gordon's  formula,  and  all  the  others  that  have  been  derived 
from  it  or  based  upon  it.  In  building  construction  this  second  element  must  not  be  omitted. 
The  metal  of  one  column  should  be  directly  over  the  metal  of  the  column  below,  continuously 
through  the  entire  height  of  the  building,  and  this  necessitates  the  application  of  the  loads  on 
the  sides  of  the  columns.  If  the  loads  are  equal  and  are  on  opposite  sides  of  the  column, 
the  effect  of  the  eccentricity  is  neutralized,  otherwise  it  increases  the  stress  on  the  side  of 
the  column  on  which  the  greater  load  is  applied.  Owing  to  the  short  length  of  most  of  the 
columns  used  in  this  construction,  and  to  the  fact  that  the  ends  are  flat  bearing,  the  value  of 

^-^  is  so  small  that  it  gives  the  third  of  these  elements  the  least  importance.  In  the  base- 
ment  columns  of  a  sixteen-story  building  the  value  of  the  term  \^/       equation  (i)  of 

Chapter  IX  is  about  .022,  while  the  value  of  the  second  term,  is  quite  commonly  as  much 
as  .07,  and  often  considerably  more.  In  the  smallest  columns  at  the  top  of  the  building  the 
value  of  the  term  (^-j  ,  owing  to  the  reduced  section,  is  about  0.220,  while  0.6  or  0.7 

vy 

would  not  be  an  unusual  value  for  the  term  —r  which  in  these  smallest  columns  occasion- 

r 

ally  doubles  the  section.  These  figures  are  taken  from  examples  at  hand.  They  show 
first  that  the  important  effects  of  eccentricity  of  loading  increase  rapidly  as  the  section  of  the 
column  decreases,  and  that  the  importance  of  this  element  in  columns  thus  eccentrically 
loaded  is  three  or  more  times  as  great  as  that  of  the  element  dependent  upon  the  flexure  of 
the  column.  These  effects  are  entirely  independent  of  the  character  of  the  column,  varying 
of  course  in  values  with  different  kinds  of  columns,  but  always  true  when  the  loading  is  as 
irregular  and  eccentric  as  the  architecture  of  modern  sixteen-  and  twenty-story  buildings 
necessitates. 

Mr.  James  Christie  in  his  report  of  tests  made  at  Pencoyd  Iron  Works,  in  a  paper  read 
before  the  American  Society  of  Civil  Engineers  in  1883,  says:  "Very  minute  changes  in  the 
position  of  the  centre  of  pressure  produces  greater  differences  in  the  resistance  of  the  bars 
than  was  anticipated."  And  in  another  place  he  says  :  "  For  reasons  not  always  evident, 
occasional  results  were  obtained  either  abnormally  high  or  low,  as  will  be  found  illustrated 
on  the  diagrams;  but  there  is  little  doubt  that  the  principal  cause  of  low  resistance  was 
eccentricity  of  axes,  or  non-coincidence  between  the  centre  of  pressure  and  the  axis  of  great- 
est resistance  of  the  specimen."    These  tests  were  made  on  small  bars,  but  the  results  would 


452 


MODERN  FRAMED  STRUCTURES. 


hold  equally  good  on  sections  and  conditions  found  in  practice.    All  that  we  have  in  actual 
experiment  bears  out  this  theory  that  seems  too  well  founded  for  question. 

There  are  two  other  factors  having  a  very  practical  bearing  on  the  strength  of  columns, 
neither  of  which  are  accounted  for  in  this  discussion  and  application  of  equation  (i)  of  Chap- 
ter IX.  One  of  these  is  contained  in  the  form  of  the  sections  used,  and  in  the  manner  in 
which  they  are  fastened  together ;  the  other  concerns  imperfections  in  workmanship  and 
material.  Both  of  these  factors  are  alike  empirical,  and  no  function  expressing  these  condi- 
tions enters  into  any  of  the  column  formulae  arranged  for  possible  practical  use.  They  are, 
however,  unlike  in  this  important  feature.  The  imperfections  of  workmanship  and  materials 
do  not  differ  greatly  with  different  kinds  of  columns,  but  rather  with  different  shops  and 
mills,  and  the  only  way  to  guard  against  them  is  to  employ  the  best  service  and  in  close  and 
careful  inspection.  On  the  other  hand,  the  form  of  the  sections  used  and  the  manner  in 
which  they  are  put  together  is  a  factor  of  strength  or  weakness  peculiar  to  each  kind  of 
column  manufactured,  and  is  an  important  feature  in  any  comparison  of  the  strength  of  dif- 
ferent kinds  of  columns. 

By  "  the  form  of  the  section  "  is  meant,  not  its  capacity  to  produce  in  the  finished  column 
a  large  moment  of  inertia  for  the  actual  area,  but  the  possible  assistance  that  the  different 
parts  of  the  section  may  afford  each  other  in  general  stiffness.  We  have  no  scientific  discus- 
sion of  this  feature  as  it  applies  to  the  strength  of  columns,  no  function  of  the  form  of  the 
sections  independent  of  their  position,  nor  of  effects  of  multiplied  punching  and  riveting  in 
any  column  formula,  and  no  adequate  comparative  tests  that  can  establish  the  relative  merits 
or  demerits  of  different  sections  in  this  respect  even  empirically.  There  is  testimony  to  the 
fact  that  it  is  an  important  consideration,  though  sometimes  authorities  do  not  agree  as  to 
the  facts  involved.  For  example,  in  a  book  published  by  the  Phoenix  Iron  Company  they 
criticise  the  Z-bar  columns  because  "  their  thin  unsupported  flanges,  flaring  out  at  extreme 
points,  are  much  to  be  deprecated,  owing  to  their  inherent  tendency  to  buckling,"  while  Mr. 
Strobel  in  his  discussion  of  the  same  column  in  a  paper  before  the  American  Society  of  Civil 
Engineers  says:  "It  will  be  seen  that  the  Z-iron  columns  compare  favorably  with  other 
columns  in  ultimate  resistance.  The  values  obtained  are  near  approximations  to  the  Water- 
town  results  with  Phoenix  columns,  and  exceed  those  heretofore  obtained  with  other  types  of 
columns.  This  favorable  showing  for  the  Z-iron  columns  should  probably  be  attributed  to 
the  fact  that  the  material  in  the  outer  periphery  of  the  cross-section,  on  which  dependence 
must  be  placed  to  hold  the  column  in  line,  is  not  weakened  by  rivet-holes,  but  is  left  solid 
and  unbroken,  and  is  therefore  in  best  shape  to  do  its  work  effectively."*  Each  of  these 
arguments,  one  for  the  strength  of  the  column  and  one  for  its  weakness,  is  based  on  the  con- 
ditions concerned  in  this  consideration  of  the  column. 

H  o  X  n  n 

Z  BARS.  PHCENIX.  LARIMER.  CHANNELS,  ANGLES  &.  PLATES. 

443. 

Fig.  443  shows  the  sections  of  the  columns  in  common  use  in  building  construction.  In 
practice  it  often  occurs  that  columns  must  be  calculated  rapidly,  and  it  is  important  to  curtail 
the  work  as  much  as  possible.  Some  of  these  column  types  are  manufactured  as  a  specialty. 
In  such  cases  the  manufacturers  have  a  working  formula  for  the  strength  of  the  columns,  and 
a  table  showing  the  safe  concentric  loads  for  columns  of  different  sizes.    These  formulae  and 


These  columns  were  made  of  iron  which  showed  very  high  elastic  limits  in  the  specimen  tests. — J.  B.  J. 


IRON  AND  STEEL  TALL  BUILDING  CONSTRUCTION. 


453 


tables  may  be  relied  upon  as  conservative.  They  are  made  for  the  use  of  unprofessional 
men,  and  it  must  be  remembered  that  they  are  for  concentric  loads  only,  that  is  to  say,  they 
represent  only  the  first  and  third  elements  of  equation  (i)  before  referred  to,  with  possibly  an 
empirical  factor  supposed  to  cover  the  conditions  of  form  and  riveting,  etc.,  peculiar  to  the. 
column.  If  a  proper  allowance  is  made  for  the  bending  moment  arising  from  the  eccentric 
loading,  and  moderately  heavy  sections  are  used  with  skilful  detailing,  there  seems  to  be  no 
good  reason  why  columns  for  most  buildings  should  not  be  subjected  to  a  higher  unit  stress 
than  is  usually  given  by  these  formulae  and  tables.  The  moment  due  to  the  eccentric  load- 
ing may  be  provided  for  as  follows:  Multiply  the  eccentric  load  by  the  distance  from  the 
point  of  its  application  to  the  centre  of  the  column ;  the  result  is  the  bending  moment  due  to 

//    fAr*  M^y 

the  eccentric  load.    Then  from  the  formula  J/.  =  —  =   ,    or    A  =     °  ',  we  find  the 

7.       J.  A 
area  of  section  required  to  resist  the  bending  moment. 

In  designing  short  columns,  as  used  for  buildings,  select  a  maximum  working  stress  on 
the  extreme  fibres  of  the  column,  on  the  side  of  the  eccentric  load,  and  neglect  all  bending  of 
the  column  for  such  working  loads.    We  then  have,  for  both  concentric  and  eccentric  loads, 

where  A  —  total  area  of  column  ;  P=  total  load  on  column,  both  eccentric  and  concentric; 
p  =  maximum  working  stress  in  lbs.  per  sq.  in. ;  M  =  bending  moment  from  eccentric  load  = 
P^v  (where      =  eccentric  load,  and  v  =  distance  of  eccentric  load  from  axis  of  column); 

y  =  distance  of  extreme  fibre  on  loaded  side  from  neutral  plane  of  column,  and  i  =  -  ;  r  = 

radius  of  gyration  of  cross-section  of  column  in  direction  of  eccentric  load.    But  for  Z-bar 

columns  and  for  most  of  the  iron  and  steel  forms     =  1.73.    If  v  be  found  in  terms  of/,  we 

may  write  v  =  ky,\  hence         -         —  ~^\'^)  •  since  y  =  i.73,^yj      3,0,  hence  our 

equation  becomes 

A  =  ^{p-^3^p.r  (II) 

This  is  an  exceedingly  simple  formula  and  readily  applied. 

479.  Wind  Bracing. — Buildings  are  always  subject  to  lateral  strains  from  wind  forces. 
But  little  attention  has  been  paid  to  this  fact  heretofore,  and  really  there  has  been  little  need. 
If  a  building  of  unusual  height  was  proposed,  an  extra  wall  was  put  into  it,  but  otherwise  the 
lateral  strength  of  the  exterior  walls  and  the  ordinary  partitions  have  almost  always  been 
quite  sufficient  to  resist  these  forces.  Where  buildings  have  been  blown  down  it  has  gener- 
ally been  shown  that  there  was  a  reckless  want  of  care  in  the  construction  where  only  a  little 
care  was  needed.  Steel  buildings,  however,  are  built  to  such  great  heights,  and  are  so  desti- 
tute of  these  ordinary  means  of  resisting  wind  forces,  that  it  is  necessary  to  give  the  subject 
much  more  serious  consideration  and  to  brace  the  steel  frames  so  that  the  strength  of  the 
buildings  in  this  respect  shall  be  assured.  This  can  be  done  in  a  variety  of  ways ;  but  the 
arrangement  of  the  rooms,  the  architectural  features,  and  other  requirements  prescribe  so 
greatly  that  the  designer  will  probably  be  left  with  but  one  way,  and  be  very  glad  that  he  has 
that  one. 

The  bracing,  whatever  it  is,  must  of  course  be  vertical,  reaching  down  to  some  solid  con. 
nection  at  the  ground.  It  should  also  be  arranged  in  some  regular  symmetrical  relation  to 
the  outlines  of  the  building.  For  example,  if  the  building  is  narrow  and  is  braced  crosswise 
with  one  system  of  bracing,  that  system  should  be  midway  between  the  ends  of  the  buildings, 
and  if  two  systems  are  used  they  should  be  equidistant  from  the  ends,  the  exact  distance 

*  This  formula  was  added  by  Prof.  Johnson  in  the  fourth  edition  of  this  work. 


4*54 


MODERN  FRAMED  STRUCTURES. 


being  unimportant,  because  the  floors,  when  finished,  are  extremely  rigid.  The  symmetrical 
arrangement  is  necessary  to  secure  an  equal  service  of  the  systems  and  prevent  any  tendency 
to  twist. 


r  ^ 

\ 

/  \ 

/ 

r  \ 

\ 

/  \ 

/ 

r  ^ 

\ 

/  \ 

/ 

\ 

/  \ 

/ 

Fig.  444. 

Fig.  444  shows  in  outline  several  ways  in  which  such  a  system  of  bracing  may  be  con- 
structed. Horizontal  lines  indicate  floors,  and  vertical  lines  indicate  columns.  It  is  obvious 
that  both  the  horizontal  iron-work  and  the  columns  must  do  a  good  part  of  the  work,  and 
that  each  arrangement  must  have  its  own  treatment  and  must  create  stresses  unlike  those 
created  by  the  other  systems.  The  loads,  however,  will  be  the  same.  If  one  system  is  used, 
the  length  of  the  building,  that  is,  the  width  of  the  side  perpendicular  to  the  direction  of  the 
bracing,  multiplied  by  the  distance  between  floors  half-way  below  and  half-way  above,  will 
equal  the  exposed  area  tributary  to  each  panel  point,  and  this  multiplied  by  the  force  per 
square  foot  will  equal  the  horizontal  external  force  applied  at  each  panel  point.  Then  the 
total  shear  at  any  point  will  equal  the  sum  of  all  the  external  forces  at  and  above  the  point 
taken.  These  shears  may  be  reduced  in  actual  practice  on  several  accounts,  and  if  such 
reduction  is  made  it  is  well  to  make  it  at  this  point  in  the  computations.  The  weight  of  the 
building  affords  some  resistance,  and  in  most  cases  is  worth  taking  into  account.  Most 
buildings  are  filled  with  tile  or  some  other  sort  of  partitions,  and  when  these  are  really 
constructed  and  their  continuance  is  assured,  there  is  no  good  reason  why  we  should  not  rely 
also  on  them  to  some  extent.  There  is  also  some  resistance  to  lateral  strains  in  the  connec- 
tion of  the  beams  to  the  columns  where  they  are  well  riveted.  Some  of  these  considerations 
will  admit  of  calculation  ;  but  in  using  them  much  must  depend  on  the  experience  and  judg- 
ment of  the  engineer. 

The  simplest  form  of  bracing  is  that  marked  a  in  Fig.  444.  In  Fig.  445,  which  is  the 
same  thing,  ^  =  the  load  or  shear  directly  tributary  to  that  panel  point;  A  =  the  sum  of 
the  loads  or  shears  tributary  to  all  the  points  above,  or,  in  other  words,  the  horizontal  com- 
ponent of  the  stress  in  rod  r\  E  —  vertical  component  of  the  stress  in  rod  r;  D  =  accumu- 
lated vertical  wind  loads  in  the  column  next  above  column  2. 

Then     A  -\-  B  —  the  horizontal  load  on  rod  s; 

{A+B)b  .    ,  ^  ^  .  ^ 
 =  vertical  component  of  the  stress  m  rod  s. 


The  compressive  stress  in  any  horizontal  strut  must  equal  A      B.    The  load  on  any 


column  2  must  equal  D-\ 


IRON  AND  STEEL  TALL  BUILDING  CONSTRUCTION. 

{A-\-B)b 


455 


,  and  this  wind  load  must  be  added  to  all  the  other  regular 


loads  on  the  column.    It  must  also  be  noted  that      ~^       is  an  eccentric  load,  the  length  of 

e 

arm  being  the  distance  from  the  bearing  or  point  of  attachment  at  the  end  of  the  horizontal 


B»»  > 


Col.  1- 


FiG.  445. 

strut  to  the  axis  of  the  column.  If  this  connection  is  to  the  axis  of  the  column  itself,  or  if 
the  rods  connect  directly  to  the  centre  of  the  column,  the  eccentricity  is  reduced  to  zero  and 
the  eccentric  load  becomes  a  dirjcl  load  the  same  as  D. 

The  regular  load  carried  1  y  column  i  resists  the  upward  vertical  component  of  the  stress 
in  rods,  connected  at  the  bottom  of  the  column,  and  the  same  is  true  at  every  other  connec- 
tion to  this  tier  of  columns.    Th:  dead  load  in  column  i  is  reduced  the  full  amount  of  the 

total  compression  for  wind  in  column  2,  that  is,  D-\-^ — '^"^  this  amount  exceeds 

the  dead  load  in  column  i  tliere  must  be  tension  in  the  connection  of  the  column  to  the  next 
one  below,  a  condition  which  is  not  provided  for  and  which  in  any  ordinary  case  should  not 
be  allowed  to  occur.  The  liorizontal  shears  multiplied  by  the  secants  will  give  the  stresses  in 
the  rods  the  same  as  in  a  truss. 

The  arrangement  marked  /;  in  Fig.  444  is  a  slight  variation  of  that  marked  a.  The 
arrangement  marked  c  consists  of  a  system  of  portals  one  above  the  other.  This  is  shown 
more  definitely  in  Fig.  446, 

A  =  accumulated  force  or  horizontal  shear  from  wind  at  the  floor  next  above  floor 

M,  applied  one  half  on  one  side  and  one  half  on  the  other  ; 
B  —  the  force  of  the  wind  or  shear  directly  tributary  to  floor  M; 


456 


MODERN  FRAMED  STRUCTURES. 


D  —  the  accumulated  vertical  wind  load  in  the  column  next  above  column  2; 
A,  B,  D  —  the  total  exterior  forces  acting  on  the  portal,  then 

{Ab  -\-  Bb  —  Bc)-^  —  vertical  resistance  due  to  A  and  B\ 


A+B 


horizontal  reactions  due  to  A  and  B. 


In  column  2  the  vertical  column  load  due  to  the  wind  must  be  added  to  the  regular  load 
of  the  column  the  same  as  in  the  arrangement  shown  in  Fig.  445.  The  load  D  and  its  equal 
reaction,  being  directly  applied  along  the  same  straight  line,  may  be  omitted  from  considera- 


( A6+B6-  Bc)i. 


Fig.  446. 

tion  in  discussing  the  strength  required  in  the  bracing,  as  may  also  the  negative  effects  equal 
to  D  which  occur  in  column  i,  the  same  as  in  the  case  shown  in  Fig.  445. 

The  horizontal  shear  along  the  line  vv  =  A  -\-  B. 

The  horizontal  shear  in  either  leg  below  the  line  vv  =  ^{A  -\-  B). 

(Ab  -\-Bb  -  Be) 

The  vertical  shear  on  all  vertical  planes  =  . 

The  thickness  of  the  web  plates  must  be  determined  by  these  shears.  It  will  be  noted 
that  the  connection  to  the  columns  must  be  equal  to  the  whole  vertical  shear.  The  direct 
compression  in  the  flange  S  —  \B.  Taking  moments  about  the  point  of  intersection  of  flange  r 
with  the  line  ww,  it  will  be  found  that  the  sum  of  the  moments  equals  zero,  that  is,  that  there 
is  no  bending  moment  in  the  portal  on  the  line  ww,  and  that  flange  /  is  not  strained  at  this 
point.  For  maximum  stress  in  flange  t  take  a  point  p  in  flange  r,  distant  x  from  the  line  iviu 
and  at  right  angles  to  any  given  section  of  the  flange     then  x  times  the  vertical  shear  divided 

X 

hy  y  ■=  the  stress  at  the  section  taken,  and  this  is  maximum  when  —  has  its  greatest  value. 

The  leg  of  the  portal  including  column  2  might  be  also  taken  as  a  cantilever  with  two  forces 

.    A-\-B        (Ab^Bb-Bc).^^  .  .         ^  ,       ,  . 

actmg  on  it,   and   ■  ,  with  flange  /  in  compression  and  the  column  itself 

acting  as  a  tension  chord.    Take  a  point  in  the  centre  of  the  column,  distant  ,r,  from  the 

A  \  B  X 

bottom  of  the  leg  and  at  right  angles  to  any  given  section  in  flange      then  — - —  .y  —  the 

X 

strain  in  flange     and  this  is  maximurn  when  —  has  its  greatest  value.    There  is  a  slight  error 


IRON  AND  STEEL  TALL  BUILDING  CONSTRUCTION. 


457 


in  this  treatment,  but  it  is  on  the  side  of  safety.  If  flange  /  has  a  section  proportioned  to 
these  maximum  stresses,  the  requirements  will  be  fulfilled. 

The  stress  and  area  required  in  flange  r  can  be  obtained  in  a  similar  manner.  The  con- 
nections of  the  portal  above  this  flange  to  the  portal  and  column  above  must  be  equal  to  \A 
at  each  leg. 

The  arrangement  marked  d  in  Fig.  444,  if  used  at  all,  would  probably  be  made  to  include 
more  than  two  columns,  and  the  stresses  would  vary  greatly  with  the  number  of  columns 
included  in  the  system.  It  is  not  an  economical  method  of  stiffening  a  structure,  as  it  pro- 
duces heavy  bending  moments  in  both  the  horizontal  struts  and  in  the  columns  themselves. 
Methods  a  and  b,  on  the  other  hand,  if  connections  are  properly  made,  do  not  cause  any 
bending  in  the  columns  or  in  the  lateral  struts. 

480.  Details. — In  many  buildings,  owing  to  irregular  lines  or  to  an  elaborate  exterior,  the 
details  of  the  construction  are  difificult  and  complicated.  Many  architects  also  have  but  little 
knowledge  of  the  practical  ways  of  connecting  and  working  metal  in  the  shop,  and  the  result 
is  that  otherwise  good  frames  are  often  decidedly  weak  in  connections  and  details.  It  is  there- 
fore very  important  that  the  general  drawings  and  specifications  should  ct  vcr  all  these  points. 

Beams  fitting  into  beams  should  have  an  eighth  of  an  inch  clearance  at  each  end. 
Standard  connections,  when  the  manufacturers  of  the  beams  have  any,  should  be  used 


3^!i  

Fig.  447. 


wherever  they  can  be  conveniently.  Care  should  always  be  exercised  to  see  that  the  flanges 
are  not  weakened  by  rivet-holes.  The  common  connection  of  a  beam  to  a  column  requires 
two  rivets  in  each  flange.  These  rivets  should  attach  'to  a  lug  directly  connected  to  the 
column  itself  by  an  equal  number  of  rivets.  As  good  a  connection,  and  in  some  ways  a  better 
one,  is  to  put  all  four  rivets  in  the  bottom  flange  and  fill  the  clearance  space  between  the 
beam  and  the  column  at  the  top  of  the  beam  with  iron  wedges  tightly  driven.  When  the 
columns  are  cut  off  under  the  beams  so  that  the  latter  can  rest  on  the  cap-plate  of  the  column 
below,  this  cap-plate  should  not  be  used  as  a  lug.    The  supports  of  brackets  in  bay-windows 


458 


MODERN  FRAMED  STRUCTURES. 


and  under  a  heavy  cornice  needs  especial  care.  These  should  be  made  as  parts  of  beams 
wherever  possible.  Fig.  447  shows  such  a  connection.  When  attached  as  shown  in  this  case, 
the  beam  marked  A  may  be  calculated  as  though  it  were  one  continuous  beam  to  the  end  of 
the  bracket.  If  the  bracket  were  attached  only  to  the  beam  marked  B,  the  latter  would  twist 
to  some  extent,  and  a  very  little  yielding  on  the  part  of  the  beam  in  this  way  would  make  the 
vertical  deflection  of  the  bracket  relatively  large.  This  would  certainly  be  an  injury  to  the 
building.    All  beams  carrying  floor  arches  should  be  provided  with  tie-rods  to  counteract  the 


SECTION  B-B  SECTION  C-O 


SECTION  A-A 


Fig.  448. 

thrust  of  the  arch.  Double  beams  should  always  be  provided  with  separators  to  equalize 
the  load.    Connections  in  high  buildings,  wherever  possible,  should  be  riveted. 

Columns  should  be  connected  to  each  other  through  the  cap-plates  by  four  or  more  rivets. 
This  should  be  independent  of  the  attachment  of  the  columns  through  the  beam  connec- 
tions. Columns  should  be  milled  at  each  end,  smooth,  and  at  right  angles  to  their  axes,  and 
^he  use  of  sheets  of  lead  or  of  any  other  rnaterial  under  one  side  of  a  column,  to  make  it 


IRON  AND  STEEL  TALL  BUILDING  CONSTRUCTION. 


459 


plumb,  is  exceedingly  bad.  The  shop-work  should  be  done  so  well  that  the  need  for  such 
adjustments  should  not  be  felt. 

Details  of  connections  in  wind  bracing  are  exceedingly  important,  as  it  is  very  easy  to 
destroy  the  efficiency  of  any  system  by  a  single  faulty  connection.  It  is  difificult  to  describe 
exactly  what  these  details  should  be,  for  the  requirements  vary  greatly  in  different  buildings. 
To  properly  detail  the  work  it  is  as  necessary  to  have  a  perfect  understanding  of  the  outward 
forces  and  the  manner  in  which  they  must  be  resisted,  as  it  is  to  calculate  the  sections 
required.  This  may  be  illustrated  in  the  connection  of  the  struts  to  the  columns  in  that 
system  of  bracing  marked  a  in  Fig.  444.  The  strut  need  not  be  connected  to  the  column 
to  resist  horizontal  forces,  for  there  is  no  force  tending  to  tear  the  strut  away  from  the 
column  in  this  direction.  The  force  to  be  resisted  here  is  vertical.  Fig.  448  shows  such  a 
connection  ;  the  strut  is  made  to  butt  the  column  squarely  instead  of  fastening  to  the  sides  of 
the  column  by  rivets  passing  through  the  two  members,  or  indirectly  through  connection 
plates,  because  the  forces  producing  stresses  in  the  bracing  at  this  point  must  come  into  the 
strut  by  compression  from  without  and  not  through  any  possible  tensile  stress.  When  the 
strut  butts  the  column  these  forces  are  introduced  into  the  strut  without  the  aid  of  rivets,  and 
the  full  value  of  all  the  rivets  can  be  used  to  resist  the  vertical  component  of  the  rod  stress. 
It  serves  also  to  keep  the  arm  at  the  end  of  the  strut,  or  the  distance  from  the  centre  of  pin  to 
the  bearing  at  the  end,  as  short  as  possible,  all  of  which  is  important.  The  top  angles  may  be 
placed  several  inches  above  the  strut,  and  a  cast  filler-block  introduced  between  them.  Such 
an  arrangement  has  several  advantages.  It  generally  happens  that  these  angles  cannot  be 
riveted  to  the  column  directly  under  the  channels  of  the  strut,  as  shown  in  Fig.  448.  The 
consequence  is  that  whatever  intervenes  must  carry  a  cross-strain.  The  cast-block  will  do  this 
well.  It  is  also  important  that  there  should  be  absolutely  no  clearance;  otherwise  the  whole 
system  would  lack  in  stiffness  and  efificiency.  The  block  can  be  cast  a  little  large,  and  if 
necessary  it  can  be  chipped  at  the  building  in  order  to  crowd  it  into  position.  The  block  also 
has  the  further  advantage  of  cheap'ness,  and  is  always  easily  obtained  ^very  detail  in  wind 
bracing  should  receive  the  most  careful  consideration. 


460 


MODERN  FRAMED  STRUCTURES. 


CHAPTER  XXIX. 
IRON  AND  STEEL  MILL-BUILDING  CONSTRUCTION* 

481.  General  Types  of  Buildings. — There  are  three  general  types  in  common  use. 
The  first  has  a  rigid  iron  frame  throughout.  Each  transverse  bent  is  made  up  as  follows :  An 
iron  roof  truss  is  supported  at  its  ends  by  iron  columns  firmly  anchored  to  masonry  piers ; 
the  bent  is  made  rigid  by  transverse  bracing,  consisting  of  knee-braces  in  intermediate  bents 
and  vertical  bracing  in  the  end  bents  between  columns.  In  the  sides  of  the  building  vertical 
bracing  is  provided  between  the  columns,  and  lateral  bracing  in  the  planes  of  the  top  and 
bottom  chords  of  the  roof  trusses.  When  long  panels  are  desired,  the  alternate  roof  trusses 
can  be  supported  on  longitudinal  girders,  or  trusses,  running  between  the  columns.  At  the 
ends  of  the  building  one  or  more  gable  columns  should  be  introduced  to  shorten  the  unsup- 
ported length  of  transverse  vertical  bracing.  The  sides  and  ends  may  be  covered  with 
corrugated  iron  sheeting  supported  on  suitable  framework. 

The  second  type  difTers  from  the  first  in  having  thin  curtain  walls  of  brick  built  up  to  the 
tops  of  the  columns  on  all  sides  of  the  building.  T  hese  walls  carry  no  vertical  loads,  but  must 
do  the  work  of  the  vertical  bracing  between  the  columns  in  the  sides  and  ends.  This  bracing 
is,  therefore,  omitted  ;  but  there  should  be  in  the  sides  of  the  building  a  top  strut  running  from 
column  to  column,  and  connecting  to  the  inner  flanges,  in  order  to  clear  the  brick  wall.  If 
desired,  the  top  strut  may  connect  on  the  centre  line  of  columns,  in  which  case  the  wall  must 
be  built  around  it. 

Buildings  of  the  first  type  of  construction  are  suitable  to  withstand  the  action  of  the 
heaviest  jib  and  travelling  cranes,  since  they  are  able  to  resist  large  horizontal  forces,  as  well 
as  the  usual  vertical  loads.  The  second  type,  while  suitable  for  buildings  of  heavy  construc- 
tion, and  especially  adapted  for  machine-shops,  are  not  so  well  fitted  to  endure  the  action  of 
cranes  of  large  capacity,  particularly  if  the  building  be  high  and  narrow.  If  for  any  reason 
the  brick  walls  are  considered  essential  and  it  is  desired  to  secure  the  greatest  strength  and 
rigidity  possible,  regardless  of  expense,  a  combination  of  these  two  types  would  give  a  result 
which  could  not  be  surpassed. 

The  last  type  of  construction  to  be  mentioned  is  well  adapted  to  buildings  intended  for 
heavy  machinery  and  light  jib-cranes,  and  may  be  used  for  suspended  travelling-cranes.  In 
this  third  type  there  are  no  iron  columns.  The  walls  of  the  building  are  brick,  and  the  roof 
trusses  rest  directly  upon  them.  Additional  rigidity  can  be  secured  by  the  introduction  of 
iron  columns  at  the  four  corners  of  the  building.  These  columns  should  be  well  anchored  to 
the  foundations.  The  roof  trusses  should  be  braced  in  the  planes  of  the  top  and  bottom 
chords.  If  there  are  no  jib-cranes  to  be  provided  for  the  bottom  chord  bracing  may  be 
put  in  alternate  panels  only ;  in  which  case,  however,  the  top  chord  bracing  should  be  in  every 
panel  to  keep  the  walls  in  line. 

482.  Vertical  Loads  or  Forces. — Under  this  head  come  the  weight  of  the  iron  frame 
and  its  covering;  the  weight  of  snow;  the  vertical  component  of  the  wind  pressure  ;  travelling 
cranes  and  their  lifted  loads;  the  weight  of  pipes,  machinery,  and  shafting;  the  action  of 


*  This  chapter  has  been  adapted  from  a  paper  contributed  to  the  Engineers'  Society  of  Western  Pennsylvania, 
October,  1892. 


IRON  AND  STEEL  MILL-BUILDING  CONSTRUCTION. 


461 


driving-belts ;  and  such  additional  loads  as  may  arise  in  individual  cases.  The  importance  of 
considering  the  action  of  stresses  due  to  cranes  is  evident  when  it  is  remembered  that 
there  are  now  in  use  jib-cranes  of  varying  capacity  up  to  fifty  tons  and  travelling  cranes  up  to 
one  hundred  tons  and  even  one  hundred  and  fifty  tons  lifting  power.  These  cranes  produce 
not  only  direct  stresses  in  main  members  and  bracing,  but  also  heavy  bending  moments  in  the 
columns  and  alternating  stresses  in  various  members  throughout  the  building.  The  action  of 
long  lines  of  shafting,  with  their  driving-pulleys  and  belts,  and  of  hydraulic  and  steam 
machinery,  frequently  contributes  a  large  share  to  the  sum  of  uncertain  and,  too  often, 
unconsidered  stresses  to  which  a  mill  building  is  subject.  While  it  is  impossible  to  determine 
these  stresses  with  accuracy,  it  is,  nevertheless,  necessary  to  make  allowance  for  them  ;  and 
the  proper  way  to  deal  with  them  is  to  assume,  as  nearly  as  may  be,  an  equivalent  uniform 
load  and  a  single  concentrated  load  which  may  be  applied  at  will  to  any  member  of  the 
structure.  These  vertical  loads  vary  so  in  amount,  depending  on  the  span  and  type  of  build- 
ing, that  it  is  impossible  to  give  an  equivalent  load  per  square  foot  which  could  be  used 
indiscriminately.  It  must  be  ascertained  independently  for  individual  cases.  The  action  of 
the  vertical  loads  on  the  structure,  when  once  they  or  their  equivalents  have  been  computed, 
is  readily  determined,  and  need  not  be  further  discussed. 

483.  Horizontal  Forces. — Under  this  head  come  the  horizontal  component  of  the  wind 
pressure  and  the  thrusts  from  the  travelling-cranes,  jib-cranes,  belts,  etc.  The  horizontal 
action  due  to  the  cranes  and  wind  is  of  considerable  importance,  and  is  an  element  that  does 
not  receive  the  attention  in  merits.  Fortunately,  however,  the  maximum  wind  and  crane 
loads  occur  but  rarely,  and  the  probability  of  their  occurring  simultaneously  is  so  small  that, 
if  we  proportion  each  member  for  the  larger  stress,  we  may  disregard  the  smaller.  Horizontal 
forces  act  on  a  building,  tending  to  move  the  structure  horizontally  and  to  overturn  it  as  a 
whole.  As  a  result,  stresses  are  produced  in  the  individual  members  throughout  the  building. 
Friction  and  the  weight  of  the  structure  are  usually  sufficient  to  counteract  the  tendency  to 
movement,  and  we  need  consider  the  action  on  the  individual  members  alone.  This  action 
can  be  best  shown  by  means  of  a  stress  diagram  (see  Plate  XXXH,  Fig.  i). 

484.  Method  of  Analysis. — The  following  conditions  will  be  assumed  to  illustrate  the 
method  of  analysis : 


Ft. 

In, 

Height  to  centre  line  of  bottom  chord  of  roof  truss  

42 

6 

16 

0 

..  7 

6 

66 

0 

64 

0 

The  building  consists  of  a  rigid  iron  framework  throughout. 

The  wind  force  will  be  taken  at  the  low  value  of  20  lbs.  per  square  foot.  The  wind  will 
be  assumed  to  blow  in  a  horizontal  direction  in  all  cases.  The  vertical  dead  load  acting  on 
roof  is  taken  at  25  lbs.  per  square  foot,  horizontal  projection  ;  with  an  additional  vertical 
loading,  acting  on  columns,  of  25  lbs.  per  square  foot,  horizontal  projection.  These  two 
loads  cover  the  weight  of  the  iron  framework  of  the  structure,  the  roof  load  being  the  weight 
of  the  trusses  and  their  covering,  and  the  additional  column  load  being  the  weight  of  the 
columns  and  their  bracing.  The  permanent  loads  only  have  been  selected  for  analysis,  so 
that  the  liability  to  alternating  wind  stresses  in  the  roof  trusses  may  be  fully  shown.  While 
the  estimate  for  dead  weights  is  made  to  more  than  cover  their  amount  in  general  practice,  no 
extreme  data  have  been  used. 

Were  we  designing  the  building  for  construction,  it  would  be  necessary  to  assume,  in  ad- 
dition to  those  already  mentioned,  vertical  loads  something  like  the  following : 


462 


MODERN  FRAMED  STRUCTURES. 


For  snow,  10  lbs.  per  square  foot. 

For  equivalent  uniform  load,  10  to  15  lbs.  per  square  foot. 
For  concentrated  load,  10,000  lbs. 

The  equivalent  uniform  and  concentrated  loads  above  mentioned  are  those  referred  to  in 
Article  482,  and  cover  the  action  of  shafting  or  other  indeterminate  loads  which  would  affect 
a  building  according  to  the  particular  purposes  for  which  it  is  intended.  If  cranes  were  to 
be  put  in  the  building,  their  action  would  also  have  to  be  considered.  For  the  present  dis- 
cussion these  loads  are  not  included  in  the  analysis. 

The  diagrams  on  Plate  I  show  : 

Case  I.  The  stresses  in  roof  trusses,  columns,  and  knee-braces,  for  wind  acting  on  one 
entire  side  of  the  building  and  roof,  for  columns  hinged  at  base. 

Case  II.  The  same  conditions  of  loading,  for  columns  rigidly  fixed  at  base  by  anchor 
bolts. 

Case  III.    The  stresses  in  roof  trusses  and  columns  for  permanent  dead  load. 

Case  IV.  The  stresses  in  roof  trusses  for  wind  on  roof  only.  The  roof  trusses  rest  on 
walls,  and  are  anchored  at  both  ends. 

For  Cases  I,  II,  and  IV  the  entire  horizontal  force  of  20  lbs.  per  square  foot  is  considered 
on  the  vertical  projection  of  all  surfaces  acted  on  by  the  wind.  The  vertical  component  is 
disregarded. 

Case  V.  The  stresses  in  roof  trusses,  columns,  and  knee-braces,  for  wind  acting  on  one 
entire  side  of  building  and  roof,  for  columns  rigidly  fixed  at  base  by  anchor  bolts. 

For  Case  V  only,  the  analysis  is  made  for  the  resultant  normal  pressure  on  the  several 
surfaces  acted  on  by  the  wind. 

It  is  customary  to  consider  the  resultant  wind  pressure  as  acting  normally  to  the  roof 
surface.  We  have,  however,  in  the  present  instance,  for  the  purpose  of  comparison,  found 
the  stresses  resulting  from  the  normal  pressure,  and  also  from  the  full  horizontal  force  of  the 
wind  acting  on  the  vertical  projection  of  the  entire  building.  In  the  latter  case  the  vertical 
component  has  been  neglected.  When  the  resultant  normal  pressure  is  taken,  as  in  Case  V, 
the  wind  stresses  may  be  found  in  one  diagram  for  this  normal  pressure,  or  the  stresses  for  the 
horizontal  and  vertical  components  may  be  found  separately  and  combined  for  total  wind 
stresses. 

Cases  II  and  V  show  a  comparison  of  the  effects  of  the  full  horizontal  force  of  the  wind 
and  of  the  resultant  normal  pressure.  All  the  conditions  in  these  two  cases  are  the  same, 
except  that  of  the  direction  of  the  resultant  wind  pressure. 

In  high  buildings,  especially  with  ventilators,  on  account  of  the  small  proportion  of  ex- 
posed roof  area,  the  difference  in  the  resulting  stresses  is  not  very  marked  ;  the  normal 
pressure  giving  in  most  members  the  smaller  results.  For  a  discussion  of  wind  pressures, 
see  Chap.  Ill,  Art.  52. 

The  intensity  of  the  wind  stresses  throughout  the  roof  truss  is  dependent,  other  conditions 
remaining  constant,  upon  the  ratio  of  the  total  length  of  the  column  to  the  length  of  the 
portion  above  the  foot  of  the  knee-brace.  The  greater  this  ratio,  the  greater  will  be  the 
stresses.  Although  the  knee-braces  have  been  assumed  of  greater  depth  than  will  ordinarily 
occur  in  practice,  an  examination  of  the  diagrams  will  show  for  Cases  I  and  II,  and  even  for 
Case  V,  that  the  wind  stresses  in  some  of  the  principal  members  on  the  windward  side  of  the 
roof  trusses  are  equal  to  the  dead  load  stresses,  while  on  the  leeward  side  the  stresses  are  of 
opposite  character  and  of  even  greater  intensity ;  note  especially  the  stresses  in  the  top  and 
bottom  chords  of  the  roof  trusses,  in  the  two  main  diagonal  ties  running  to  the  ridge,  in  the 
knee-braces  and  in  the  adjoining  diagonals. 

The  horizontal  reaction  or  shear  at  the  base  of  the  columns  is  a  known  quantity;  it  re- 
mains the  same  whether  the  columns  are  hinged  or  rigidly  fixed  at  the  piers.    For  the  two 


IRON  AND  STEEL  MILL-BUILDING  CONSTRUCTION. 


465 


columns  of  a  bent  it  is  equal  to  the  horizontal  component  of  the  wind  on  one  panel  of  the 
building  and  roof.  This  total  shear  or  reaction  is  assumed  to  be  equally  distributed  between 
the  two  columns.  The  wind  loads  are  considered  concentrated,  and  the  vertical  reactions  at 
the  feet  of  the  columns  are  disregarded  in  the  analysis  of  stresses  in  trusses  and  columns, 
but  are  considered  in  the  calculation  for  anchor  bolts,  masonry,  etc. 

External  Forces. — The  columns,  in  deflecting  from  the  wind  loads,  have  a  leverage  action 
producing  certain  horizontal  reactions  at  the  feet  of  the  knee-braces,  and  at  the  bottom  chords 
of  roof  trusses,  which  must  be  considered  as  external  forces  in  finding  the  vertical  reactions, 
and  in  the  analysis  for  stresses  in  the  knee-braces  and  roof  truss. 

Vertical  Reactions  from  Horizontal  Forces. — -Taking  the  centre  of  moments  at  any  point 
in  the  axis  of  either  column,  the  vertical  reaction  at  the  opposite  column,  in  any  case,  is  equal 
to  the  algebraic  sum  of  the  moments  of  all  the  external  forces  above  this  point,  divided  by  the 
span  of  the  building.  The  centre  of  moments  may  be  taken  at  any  point  in  the  axis  of  the 
column ;  but  the  most  convenient  points  are  at  the  bases  of  the  columns,  at  the  points  of  con- 
traflexure,  and  at  the  feet  of  the  knee-braces. 

Vertical  Reactions  from  Vertical  Forces. — The  vertical  reaction  at  either  column  resulting 
from  the  vertical  components  of  the  horizontal  wind  force  is  equal  to  the  sum  of  the  moments 
of  these  vertical  forces  about  a  point  in  the  axis  of  the  other  column  divided  by  the  span  of 
the  building. 

For  analyses  in  which  only  the  horizontal  component  of  the  wind  force  is  dealt  with,  as 
in  Cases  I  and  II,  the  vertical  reaction;:  resulting  from  the  horizontal  component  only  need  be 
considered.  When,  however,  as  in  Case  V,  the  analysis  is  made  for  the  horizontal  and  vertical 
components  of  the  resultant  normal  wind  pressure,  the  algebraic  sum  of  the  corresponding 
vertical  reactions  must  be  considered.  In  all  cases  the  wind  and  dead  load  stresses  have  been 
determined  separately,  for  the  purpose  of  comparison  and  to  locate  the  alternating  stresses. 
To  determine  the  net  stresses  in  the  members  and  the  resultant  overturning  action  on  the 
building,  the  stresses  and  reactions  for  dead  and  wind  loads  must  be  combined  algebraically. 

485.  Action  of  Horizontal  Forces  on  the  Columns. — The  column  in  resisting  hori- 
zontal forces  acts  as  a  beam,  and  when  rigidly  anchored  it  may  be  considered  as  fixed  in 
direction  at  the  base,  but  when  not  properly  anchored  it  must  be  considered  as  hinged  at  the 
base,  or,  in  other  words,  free  to  rock  on  the  pier.  Theoretically  we  might  treat  the  column  as 
fixed  in  direction  at  the  top  also,  when  rigidly  connected  to  the  roof  truss  ;  but  as  we  are  not 
at  all  sure  of  realizing  a  sufificient  degree  of  rigidity  at  this  point,  it  will  be  safer  to  consider 
the  column  as  simply  supported  at  the  top  in  all  cases.  In  reality  there  would  be  but  little 
gain  in  fixing  the  column  in  direction  at  the  top,  as  it  must  be  done  at  the  expense  of  result- 
ing bending  moment  in  the  roof  truss.  The  column,  then,  in  resisting  horizontal  forces  will, 
when  rigidly  anchored  at  the  base,  act  as  a  beam  fixed  at  one  end  and  supported  at  the  other, 
but  when  not  properly  anchored  it  will  act  as  a  simple  beam  supported  at  both  ends.  The 
anchor  bolts  which  fix  the  column  have  to  resist  the  bending  moment  at  the  base ;  or,  in 
other  words,  they  must  counteract  the  tendency  of  the  column  to  rock  on  the  pier.  They 
must  also  resist  the  horizontal  thrusts  not  overcome  by  friction.  The  rocking  tendency  is 
influenced  by  the  width  of  colunm  base  and  by  the  depth  of  vertical  bracing.  The  lower  the 
bracing  extends,  the  smaller  will  be  the  bending  moment  at  the  foot  of  the  column,  and  the 
less  the  pull  on  the  anchor  bolts.  If  the  bracing  were  to  extend  the  entire  length  of  the 
column,  the  bending  moment  at  the  base  and  the  pull  on  the  anchor  bolts  would  be  reduced 
to  zero.  In  practice  the  bracing  can  run  down  the  column  a  short  distance  only,  and  bending 
moments  will  occur  at  the  foot  of  the  bracing  and  at  the  base  of  the  column.  As  a  result, 
the  column,  acting  as  a  beam,  will  be  distorted,  and  a  point  of  contraflexure  will  occur  in  the 
unsupported  portion.    If  the  anchor  bolts  be  removed,  the  case  will  be  a  beam  supported  at 


464 


MODERN  FRAMED  STRUCTURES. 


both  ends.  The  point  of  contraflexure  will  then  disappear  and  the  bending  will  be  largely 
increased. 

From  this  discussion  we  observe  the  following  important  facts :  Deep  bracing  and  strong 
anchorage  reduce  the  stresses  in  the  columns,  roof  trusses,  and  the  bracing  itself.  In  design- 
ing the  column  the  moments  at  the  base  and  at  the  connection  of  the  bracing  should  be  pro- 
vided for  in  the  section.  The  anchor  bolts  should  be  made  of  sufficient  section  to  thoroughly 
fix  the  column  at  its  base. 

The  analysis  for  Case  I  is  the  same  as  that  for  similar  cases  in  Art.  115,  of  Chap.  VII,  so 
far  as  the  reactions  at  the  base  of  the  columns  are  concerned.  The  analysis  of  Case  II  is 
given  in  Art.  151,  Chap.  X.  In  this  case  the  horizontal  and  vertical  reactions  act  at  the  point 
of  inflection,  as  found  in  Art.  151,  and  the  remaining  analysis  is  the  same  as  that  in  Art.  115. 
In  the  analysis  for  these  reactions  as  shown  in  Plate  XXXII,  the  tops  of  the  posts  were 
assumed  to  remain  in  the  same  vertical  with  the  bottom,  and  the  ordinary  continuous  girder 
formulae  were  used,  as  is  graphically  shown  in  Fig.  449.    This  puts  the  point  of  inflection  too 


Xoadlng 


Uoments 


-M 


S2 


1"^ 


Shears 


Case  n. 

Fig.  449. 


low,  and  increases  the  stresses  in  the  knee-braces  and  roof  truss.  The  more  rigid  analysis 
given  in  Art.  151  should  be  used. 

After  the  horizontal  and  vertical  reactions  have  been  determined,  the  stresses  in  the  roof 
trusses  and  the  knee-braces  can  be  easily  found  graphically.  The  graphical  treatment  is  shown 
in  full  on  Plate  XXXII,  Fig.  2.  Taking  thejoot  of  the  windward  knee-brace  as  a  convenient 
starting-point,  the  force  polygon  is  constructed  as  follows  :  from  « to  /to  c-d-e-f-x-y-t  to  a  to  close 
at  point  of  starting;  the  polygon  being  made  up  of  the  wind  loads  and  the  horizontal  and 
vertical  reactions  mentioned  above.  The  shear  at  the  base  of  the  column  does  not  directly 
appear  in  the  force  polygon  or  in  the  stress  diagram,  but  its  equivalent  has  been  considered  in 
the  load  at  the  foot  of  the  knee-brace  and  in  the  shear  at  the  top  of  the  column. 

Notice  that  at  the  foot  of  the  windward  knee-brace  the  concentrated  wind  load  at  that 
point  must  be  deducted  from  the  load  /*,  or  to  find  the  net  horizontal  reaction  which  is  to 
be  used  in  the  force  polygon  and  which  is  the  horizontal  component  of  the  stress  in  the  wind- 
ward knee-brace. 

The  analysis  for  the  horizontal  action  of  crane  loads  is  similar  to  that  given  for  wind, 
except  that  it  is  simpler,  as  there  are  fewer  loads  to  deal  with ;  therefore  it  is  not  necessary 


IRON  AND  STEEL  MILL-BUILDING  CONSTRUCTION. 


to  give  the  analysis  for  this  class  of  forces,  but  in  practice  the  maximum  stresses  arising  from 
either  wind  or  crane  loads  should  be  provided  for,  considering  such  part  of  the  crane  load  as 
may  be  deemed  proper  for  the  bent  under  consideration. 

486.  Systems  of  Bracing. — Mill  buildings  should  be  thoroughly  braced  to  resist  the 
action  of  horizontal  forces  and  to  keep  them  properly  lined  up.  For  this  purpose  longitu- 
dinal and  transverse  bracing  in  a  vertical  plane  and  lateral  bracing  in  a  horizontal  plane  are 
required. 

Ventilator  Bracing. — There  should  be  a  strut  running  from  the  top  of  each  ventilator 
post  to  the  ridge  of  the  main  roof  truss,  to  prevent  distortion  of  the  ventilator  frame.  Ver- 
tical longitudinal  bracing  is  needed  in  the  sides,  and  at  the  centre  line  of  ventilators,  to  keep 
the  frames  upright  and  parallel.  When  swinging  windows  are  placed  in  sides  of  ventilators 
the  bracing  there  must  be  omitted,  and  replaced  by  stiff  bracing  fastened  to  the  ventilator 
purlins. 

Rafter  Bracing. — The  purpose  of  this  bracing  is  to  keep  the  trusses  from  overturning, 
and  it  should  be  strong  enough  to  resist  the  action  of  the  wind  on  the  gable  end  of  the  roof. 
This  bracing  may  be  omitted  in  alternate  bays. 

Bottom  Chord  Bracing. — The  purpose  of  this  bracing  is  to  keep  the  columns  in  line  and 
to  distribute  concentrated  loads. 

Vertical  Bracing  between  Columns  should  be  provided  to  transmit  to  the  foundations  all 
stresses  due  to  horizontal  forces.  The  greater  the  depth  of  this  bracing  for  given  length  of 
column,  the  less  will  be  the  resulting  stresses  in  the  columns,  trusses,  and  in  the  bracing  itself. 
Usually  it  may  come  down  to  a  point  eight  to  ten  feet  from  the  ground.  The  bracing  in  the 
sides  takes  the  longitudinal  thrust  of  the  wind  and  crane  loads,  and  the  bracing  in  the  ends 
resists  the  part  of  the  transverse  action  of  those  loads  received  through  the  bottom  chord 
bracing. 

It  will  be  well  to  consider,  in  a  general  way,  the  action  of  the  wind  and  crane  loads  on 
these  systems  of  bracing  in  the  sides  and  ends  of  a  building.  Both  of  these  classes  of  forces, 
if  allowed,  would  act  on  the  structure  in  the  same  general  manner ;  but  the  different  condi- 
tions under  Avhich  they  are  applied  warrant  the  provision  of  separate  systems  of  bracing  to 
transmit  them  to  the  foundations.  It  is  impracticable,  in  long  buildings,  to  make  the  bottom 
chord  bracing  heavy  enough  to  carry  the  cumulative  wind  loads  from  the  successive  panels  to 
the  ends  of  buildings.  It  is,  therefore,  necessary  that  each  intermediate  bent  should  resist  its 
own  wind  load  and  a  portion  of  the  distributed  crane  loads.  To  do  this  knee-braces  are  pro- 
vided at  each  bent,  running  from  bottom  chord  to  column.  The  conditions  of  head  room 
allow  the  knee-brace  to  extend  down  the  column  a  short  distance  only,  and  the  resulting 
stresses  from  the  wind,  as  has  been  shown  in  the  diagram,  are  large  in  the  trusses  and  columns 
and  in  the  knee-braces  themselves.  These  members  should,  therefore,  be  made  abundantly 
strong  to  resist  these  stresses.  The  uniform  wind  loads  having  been  provided  for,  the  crane 
loads  and  other  local  loading  only  remain  to  be  considered.  The  bottom  chord  bracing  must 
take  these  loads  and  distribute  them  to  the  adjacent  bents,  transmitting  the  residue,  if  any, 
to  the  vertical  bracing  in  the  ends  of  the  building.  This  is  an  economical  arrangement,  since 
the  maximum  crane  stresses  occur  simultaneously  in  two  or  three  panels  only,  and,  conse- 
quently, are  not  seriously  cumulative.  Crane  stresses  are  occurring  constantly,  and  their 
repeated  action  would  be  a  severe  test  upon  the  rigidity  of  the  structure  if  each  bent  were 
required  to  resist  the  stresses  from  loads  in  the  adjacent  panels.  On  the  other  hand,  large 
wind  loads  occur  rarely,  and  the  individual  bent  can  withstand  them  without  injury.  We  do 
not  mean  to  say  that  in  practice  the  wind  will  be  taken  by  the  individual  bents  and  the  crane 
loads  by  the  bottom  chord  bracing,  because  there  is,  necessarily,  some  uncertainty  as  to  the 
distribution  of  these  loads;  but  the  bottom  chord  bracing  will  have  served  its  purpose  if  it 
relieve  individual  bents  of  the  racking  effect  of  concentrated  loads. 


466 


MODERN  FRAMED  STRUCTURES. 


487.  Types  of  Roof  Trusses. — Sections  and  Details. — While  there  are  several  types  of 
roof  trusses  well  adapted  to  mill  buildings,  probably  the  one  in  most  common  use  is  the 
"  French  "  roof  truss.  This  truss  is  simple  in  design  and  suitable  for  either  light  or  heavy 
construction.  Whatever  the  type,  it  is  always  best  to  make  the  bottom  chord  of  the  truss 
straight,  except  when  the  conditions  of  head  room  require  it  to  be  raised. 

Irrespective  of  the  type  of  roof  truss,  there  are  two  radically  different  styles  of  con- 
struction which  may  be  used :  one  is  pin-connected  and  the  other  is  riveted  work  throughout. 
While  there  is  a  large  variety  of  sections  from  which  to  choose  in  the  riveted  style,  the 
T-shaped  section  is  most  commonly  used.  This  is  composed  of  two  angles  placed  back  to 
back,  with  or  without  a  single  web  plate  between.  This  form  of  section  may  be  used  for 
both  tension  and  compression  members.  A  single  flat  bar  may  be  used  for  the  smaller 
tension  members.  Roof  trusses  built  in  this  manner  lack  somewhat  in  rigidity  and  lateral 
stiffness. 

In  the  all-iron  type  of  building — that  is,  where  the  roof  trusses  rest  directly  upon  the 
columns — rigidity  and  stiffness  are  important  factors,  and  the  construction  shown  in  Fig.  I. 
Plate  XXXIII,  is  frequently  used  with  satisfactory  results.  In  this  case  the  compression  mem- 
bers are  made  up  of  two  channel-bars  turned  back  to  back  and  latticed,  forming  a  box  section, 
For  members  subject  to  tension  only  loop-eye  rods  are  used.  The  truss  is  pin-connected 
throughout.  Upon  examination  of  the  figure,  it  will  be  observed  that  all  the  connections  are 
readily  made,  and  that  the  details  are  simple  in  design.  But  little  riveting  is  required,  the 
shop-work  can  be  easily  and  quickly  turned  out,  and  the  building  rapidly  erected  in  the  field. 
The  bottom  chords  of  the  trusses  will  stand  heavy  loads  in  bending,  and  afford  especially  good 
connections  for  jib-cranes  and  lateral  bracing.  The  roof  trusses  have  such  strength  and  rigidity 
as  to  impart  stiffness  to  the  entire  building.  This  style  of  construction  is  well  adapted  to 
both  light  and  heavy  buildings. 

There  is  another  style  of  construction  suitable  for  heavy  work  only,  which  is  quite  as 
rigid  as  the  one  just  mentioned.  In  this  the  top  and  bottom  chords  are  of  box  sections,  and 
made  up  of  plates  and  angles  instead  of  channels.  The  web  members  have  an  I  shape,  and 
are  made  up  of  four  angles  placed  back  to  back  in  pairs  and  latticed.  All  the  truss  members 
are  stiff  and  may  have  either  pin  or  riveted  connections. 

When  these  built  sections  are  used  with  riveted  connections,  the  shop-work  is  somewhat 
less  expensive  than  for  the  pin-connected  channel  construction.  When,  however,  pin  connec- 
tions are  to  be  used  for  both  styles  of  construction,  the  channel  sections  require  less  shop-work 
and  afford  simpler  details  than  the  built  sections. 

Roof  columns  and  light  crane  columns  may  be  constructed  of  Z  bars,  channels,  or  plates 
and  angles,  but  for  simplicity  of  details  the  channel  columns  are  to  be  preferred  to  all  others. 
For  heavy  crane  columns,  however,  it  may  be  necessary  to  use  plates  and  angles  to  secure  a 
column  of  sufficient  width. 

Referring  again  to  roof  trusses,  whatever  style  of  construction  is  used,  the  following  points 
should  be  well  considered  :  The  trusses  must  be  designed  to  withstand  the  maximum  stresses 
produced  by  any  possible  combination  of  vertical  and  horizontal  loads  or  by  the  vertical  loads 
alone.  The  bottom  chords  of  trusses  should  be  made  stiff  enough  to  take  both  the  tension 
and  compression  to  which  they  are  subjected  as  well  as  the  bending  from  local  loads.  They 
must  afford  a  convenient  connection  for  the  longitudinal  struts  in  the  bottom  chord  bracing; 
in  fact,  they  are  themselves  important  struts  in  that  system  of  bracing. 

488.  Columns  and  Girders  for  Travelling-cranes. — Track  girders  for  light  travelling- 
cranes  may  be  supported  on  brackets  on  the  roof  columns ;  but  as  this  arrangement  gives  an 
eccentric  loading,  it  is  better,  for  heavy  cranes,  to  provide  a  separate  column  to  carry  this  load. 
To  prevent  unequal  settlement,  the  crane  columns  and  roof  columns  should  stand  on  the  same 
base.    The  connection  between  these  columns  should  be  rigid,  so  that,  as  far  as  possible,  they 


IRON  AND  STEEL  MILL-BUILDING  CONSTRUCTION. 


467 


may  act  as  one  member.  The  girders  must  be  thoroughly  connected  to  the  crane  columns 
and. braced  laterally  to  the  roof  columns.  Knee-braces  from  crane  columns  to  girders  give 
longitudinal  stiffness.  For  track  girders  I  beams  or  plate  girders  are  preferable  to  lattice  gir- 
ders. Long,  heavy  girders  should  have  the  box  section,  with  diaphragms  at  intervals  between 
the  webs.    (See  Plate  XXXIII,  Fig.  5.) 

Rails  for  Track. — In  order  that  the  alignment  of  the  track  may  be  true,  it  is  best  to  drill 
the  holes  for  rail-fastening  with  girders  in  position  and  the  rails  lined  up.  Oak  packing  is  fre- 
quently put  under  the  rails  and  at  the  girder  seats  to  insure  smoothness  in  the  running  of  the 
cranes.  Several  rail  sections,  with  method  of  fastening,  are  shown  on  Plate  XXXII.  Details 
of  crane  columns  and  track  girders  are  shown  on  Plate  XXXIII,  Fig.  5. 

489.  Details  of  Construction. — Splices  and  Connections. — Members  subject  to  tension 
and  compression  should  have  their  splices  and  end  connections  made  to  resist  the  maximum 
stresses  of  both  kinds.  When  bending  stresses  occur,  they  should  be  considered  in  designing 
the  splices ;  this  necessitates  both  flange  and  web  splices.  As  an  example,  the  splices  in  the 
bottom  chord  of  the  I'oof  trusses  should  be  able  to  develop  the  full  strength  of  that  member 
in  tension,  compression,  and  bending.  The  splice  at  the  ridge  of  the  roof  truss  has  to  resist 
the  horizontal  thrust  from  the  rafters  and  any  vertical  shear  which  may  arise  from  unsym- 
metrical  loading.  It  is  well  to  remember  that  the  wind  is  an  important  case  of  unsymmetrical 
loading.  In  addition,  provision  must  be  made  to  introduce  into  the  rafters  the  stresses  from 
the  v/eb  members  which  meet  at  this  point. 

The  connection  of  roof  trusses  to  columns  has  to  resist  large  horizontal  and  vertical 
stresses.  In  the  analysis  given  for  wind  in  Case  I,  shown  on  Plate  XXXII,  the  horizontal  shear 
on  the  leeward  side  is  40,000  pounds  and  the  upward  pull  from  the  roof  truss  is  22,000  pounds, 
which  gives  a  resultant  of  about  46,000  pounds  to  be  resisted  by  the  connection.  This 
resultant  is  net,  that  is,  exclusive  of  the  25,000  pounds  dead  load  from  the  rafter.  On  the 
windward  side  there  occurs  a  resultant  of  55,000  pounds  from  wind  alone,  or  70,000  pounds 
from  wind  and  dead  loads  combined.  The  style  of  connection  determines  whether,  on  the 
windward  side,  the  larger  or  the  smaller  resultant  should  be  provided  for. 

Cohunn  Bases. — The  gusset  plates  and  bracket  angles  at  the  foot  of  the  column  have 
several  important  duties  to  perform,  which  require  them  to  be  thoroughly  connected  thereto. 
They  assist  in  distributing  the  column  loads  to  the  base-plate  ;  they  give  stability  to  the 
column  by  broadening  its  base,  and  furnish  a  connection  for  the  anchor  bolts.  The  base- 
plate should  be  large  and  strong  enough  to  distribute  to  the  pier  the  loads  received  from  the 
enlarged  column  base.  The  anchor  bolts  should  pass  through  the  bracket  angle,  and  at  the 
same  time  be  placed  as  far  from  the  centre  line  of  the  column  as  practicable.*  They  should 
be  anchored  in  the  masonry  sufficiently  to  develop  the  full  strength  of  the  bolts.  Large 
anchor  bolts  should  be  built  deeply  into  the  masonry  ;  small  anchor  bolts,  however,  may  be 
roughened  or  split  and  wedged  and  set  in  drilled  holes,  with  hydraulic  cement,  lead,  or  sul- 
phur.   (See  Plate  XXXII,  Fig.  i.) 

Piers. — The  masonry  piers  for  supporting  the  columns  should  be  of  sufficient  dimensions, 
weight,  and  strength  to  resist  the  vertical  loads,  the  horizontal  thrust,  and  the  overturning 
action  from  the  column.  They  should  extend  well  below  the  frost  line  and  reach  a  firm 
bottom.  If  necessary,  concrete  or  piles  ma\'  be  used  to  secure  a  suitable  foundation.  To 
secure  a  uniform  distribution  of  the  column  loads,  the  piers  should  be  provided  with  a  cap 
consisting  of  either  a  single  stone  or  a  cast-iron  plate.  If  the  stone  be  used,  it  should  be  large 
enough  to  give  a  margin  all  around  the  column  base  equal  to  one  third  of  its  thickness,  which 
latter  should  be  at  least  one  third  its  largest  dimension. 

*  This  connection  cannot  possibly  develop  the  full  bending  resistance  of  the  colunnn.  Vertical  angles  riveted  to 
the  base  of  the  column,  with  a  top  plate  to  receive  the  nut  of  the  bolt,  is  to  be  preferred  if  the  column  is  assumed  to  be 
fixed  in  position  at  bottom.    (See  Figs.  426  and  434.) — J.  B.  J. 


468 


MODERN  FRAMED  STRUCTURES. 


Purlins. — Angles,  Z  bars,  or  I  beams  may  be  used  for  roof  purlins,  with  or  without  trussing. 
Single  channels  do  not  have  sufificient  lateral  stiffness  to  make  good  purlins.  Several  styles 
of  trussed  purlins  are  shown  on  Plate  XXXVII,  Figs,  i,  2,  and  3.  Fig.  2  shows  a  simple  and 
desirable  style  of  trussing,  using  star  shapes  and  flat  bars.  Fig.  i  shows  a  good  section  for 
a  long  and  heavily  loaded  purlin,  made  up  of  two  channels,  forming  a  box  section  and  held 
together  by  tie-plates.  Loop-eye  rods  are  used  in  the  trussing,  which  is  pin-connected.  Care 
.should  always  be  taken  to  turn  the  shape  used  as  a  purlin  so  as  to  secure  the  greatest  vertical 
depth.  Figs.  5  and  6  show  respectively  the  correct  and  incorrect  method  of  placing  the 
purlin  upon  the  rafter.  Fig.  5  also  shows  the  method  of  bracing  I-beam  purlins  to  rafter  by 
the  use  of  bent  plates.  Purlins  act  as  longitudinal  struts  in  the  system  of  rafter  bracing.  At 
the  ridge,  under  ventilators,  no  purlins  are  required.  It  is,  therefore,  necessary  to  have  a  ridge 
strut  to  complete  this  system  of  bracing.  Long  purlins  are  liable  to  sag  in  the  plane  of  the 
roof.  Fig.  4  shows  a  method  of  holding  them  in  place  by  tie-rods  between  the  purlins  and 
running  from  ridge  to  eaves. 

Expansion. — Roof  trusses  resting  on  brick  walls  should  have  one  end  free  for  expansion. 
A  wall  plate,  for  a  sliding  surface,  should  be  used  for  spans  to  about  75  feet,  and  rollers  should 
be  provided  for  larger  spans.  Roof  trusses  resting  on  columns  must  be  attached  rigidly 
thereto,  no  provision  being  made  for  expansion.  It  is  customary  to  introduce  an  expansion 
panel  in  long  buildings,  at  points  100  to  150  feet  apart,  to  provide  for  longitudinal  expansion. 

Corrtigated  Iron  SJieeting. — The  use  of  corrugated  sheeting  to  close  the  sides  and  ends  of 
buildings,  where  brick  walls  are  not  present,  has  already  been  mentioned.  It  is  used  as  well 
for  a  roof-covering,  and  has  the  advantage  of  being  cheap,  light  in  weight,  and  incombustible. 
Furthermore,  it  is  quickly  and  easily  put  on  and  readily  renewed.  Its  most  objectionable 
quality  is  its  liability  to  rust.  This  can  be  retarded  if  the  sheeting  is  kept  well  painted  on 
both  sides,  and  still  further  by  using  galvanized  sheeting.  Common  weights  of  sheeting  for 
roofs  are  Nos.  18  and  20,  and  for  sides  of  buildings  Nos.  20and  22.  For  No.  20  sheeting,  pur- 
lins should  be  spaced  not  over  6  feet  centres,  and  preferably  less  than  this. 

Lighting  and  Ventilation. — Ample  provision  should  be  made  for  lighting  and  ventilation. 
Windows  with  swinging,  sliding,  or  fixed  sash  may  be  introduced  in  the  sides  and  ends  of  the 
building  and  in  the  sides  of  ventilators.  Louvres  for  ventilation  may  replace  the  windows 
where  desired.  Additional  lighting  surface  can  be  obtained  by  the  introduction  of  skylights 
on  the  main  and  ventilator  roofs.  In  the  sides  of  a  building  which  has  brick  walls  it  is  a 
simple  matter  to  provide  openings  for  windows  and  doors.  In  buildings  with  no  brick  walls, 
framework  between  the  columns  is  necessary  to  support  the  doors,  windows,  and  sheeting. 
This  framework  may  be  all  timber,  all  iron,  or  a  combination  of  iron  and  timber.  The  frame- 
work entirely  of  timber  is  suitable  for  light  buildings  only.  The  framework  entirely  of  iron  is 
required  in  buildings  where  combustible  material  is  prohibited.  In  the  latter  the  bracing  and 
framework  supporting  the  windows  are  of  structural  iron,  and  the  window  frames  and  sash  are 
galvanized  iron.  An  all-iron  frame  for  a  gable  is  shown  on  Plate  XXXVII,  Fig.  7.  The  com- 
bination timber  and  iron  framework  is  suitable  for  all  buildings  where  absolute  fire-proof  con- 
struction is  not  needed.  In  this  case  we  have  a  complete  system  of  iron  bracing  between 
the  columns.  The  main  girts  are  of  iron,  but  the  girts  and  posts  supporting  the  windows  and 
corrugated  iron  are  of  timber  and  are  introduced  between  the  main  girts  wherever  needed. 
This  style  of  construction  is  shown  on  Plate  XXXIII.  In  the  all-iron  type  of  construction  the 
bracing  between  the  columns  is  frequently  stopped  8  to  12  feet  above  the  ground.  The  space 
below  the  bracing  may  be  left  entirely  open  or  closed  with  corrugated  sheeting  and  swinging, 
sliding,  or  lifting  doors. 

General  Conclusions. — In  the  design  of  mill  buildings  of  the  present  day,  it  seems  to  the 
writer  that  there  are  certain  stresses  incident  to  the  every-day  operation  of  the  mill  which  do 
Qot  receive  proper  recognition.    We  can  with  profit  make  a  comparison  of  the  design  of  the 


IRON  AND  STEEL  MILL-B UILDTNG  CONSTRUCTION. 


469 


mill  building  with  that  of  the  railroad  bridge,  since  in  general  the  same  principles  apply  to 
both.  In  the  latter,  provision  is  made  for  certain  secondary  stresses,  such  as  impact,  wind, 
and  centrifugal  force.  In  addition,  the  probable  increase  in  loading  during  the  life  of  a  struc- 
ture is  considered.  In  like  manner,  we  have  in  the  mill  certain  secondary  stresses  due  to  the 
action  of  cranes  and  wind,  of  equal  importance.  The  question  of  the  future  increase  in 
loading  of  the  mill  building  should  be  given  the  same  attention  as  it  receives  in  bridge 
construction.  We  have  already  considered  the  action  of  these  secondary  forces  upon  the 
building,  and  have  shown  that  they  must  be  provided  for  by  the  use  of  liberal  sections  and 
suitable  bracing  in  order  to  secure  rigidity.  The  uncertainty  respecting  the  increase  in  loading 
which  future  conditions  may  dictate  for  the  structure  is  especially  marked  in  the  case  of  ex- 
tensive plants,  in  which,  on  account  of  rapid  development,  radical  changes  are  not  unusual. 
While  the  writer  would  not  be  understood  to  advocate  the  introduction  of  material  in  members 
throughout  the  structure  regardless  of  their  present  or  probable  future  requirements,  he  does 
hold  that  in  the  long-run  it  is  economy,  in  the  case  of  permanent  structures,  to  provide  not 
only  for  loads  which  it  is  known  will  occur,  but  for  those  which  experience  teaches  are  within 
a  reasonable  range  of  probability.  The  objection  to  all  this  is  that  it  costs  money,  but  small 
first  cost  is  not  always  true  economy.  A  cheap  building  will  in  time  cost  enough  for 
repairs  and  remodelling  to  make  it  an  expensive  investment.  Furthermore,  delays  in  the 
operations  of  the  mill,  which  are  liable  to  result  from  weak  and  faulty  construction,  are  at 
best  expensive  drawbacks,  and  frequently  are  far-reaching  in  their  consequences. 

A  principle  well  worth  noting,  and  one  which  argues  strongly  in  favor  of  designing  for 
liberal  loads,  may  be  stated  in  this  connection  :  After  the  material  has  been  provided  to  make 
a  structure  strong  enough  to  carry  a  moderate  loading,  the  introduction  of  a  reasonable 
amount  of  extra  material  will  give  an  increase  in  carrying  capacity  which  is  entirely  out  of 
proportion  to  the  expense  incurred  for  such  increase. 

In  conclusion,  we  may  say  that  the  three  most  essential  factors  in  the  design  and  con- 
struction of  mill  buildings,  named  in  the  order  of  their  importance,  are  strength,  simplicity, 
and  economy. 

The  accompanying  plates  (XXXII  to  XXXVII)  represent  several  t}i)cs  of  structures 
recently  erected  in  the  vicinity  of  Pittsburg.*  They  are  largely  self-explanatory  and  need  no 
further  description  in  the  text. 

ANALYTICAL  TREATMENT. 

490-  Analysis  for  Wind.- — The  following  is  a  brief  statement  of  the  conditions  assumed 
and  the  methods  employed  in  the  analysis  for  wind  stresses.  The  reactions,  shears,  and 
moments  are  determined  analytically;  the  resulting  stresses  in  the  knee-braces  and  roof  trusses 
are  determined  graphically,  as  shown  on  Plate  XXXII. 

Case  I  in  the  accompanying  sketch  represents  the  loads,  shears,  and  moments  for  column 
hinged  at  base  and  acting  as  a  simple  beam  supported  at  both  ends. 

Case  II  represents  the  loads,  shears,  and  moments  for  column  rigidly  fixed  in  direction  at 
base  by  anchor  bolts  and  acting  as  a  beam  fixed  at  one  end  and  supported  at  the  other. 

NOTATION. 

Known  Terms : 

b  —  span  of  roof ; 

/=  length  of  beam  or  column  ; 

a  —  distance  from  base  of  column  to  foot  of  knee-brace  or  to  the  point  of  application 
of  the  load  P,  or  ; 


*  The  authors  are  indebted  to  the  courtesy  of  the  management  of  the  Keystone  Bridge  Works  for  the  drawings  of 
work,  constructed  by  them,  from  which  these  details  were  selected. 


47C 


MODERN  FRAMED  STRUCTURES. 


I 

S,  =  the  horizontal  component  of  reaction,  or  shear,  at  base  of  column  when  hinged  at 
base.    Case  I  ; 

5,  =  horizontal  component  of  reaction  or  shear  at  base  of  column  when  fixed  at  base- 
Case  II  ; 

6',  =  ^3  =  one  half  external  horizontal  force. 


Case  I.  Case  n. 


Fig.  450. 

Unknoivn  Terms  : 

S,'  =  shear  at  top  of  column  when  fixed  at  base ; 

P,  =  horizontal  thrust  or  load  at  foot  of  knee-brace  due  to  the  leverage  action  of  the 
column  when  hinged  at  base  ; 
=  horizontal  thrust  or  load  at  foot  of  knee-brace  due  to  the  leverage  action  of  the 

column  when  fixed  at  base; 
=  bending  moment  at  the  foot  of  column  when  fixed  at  base; 

—  moment  at  any  section  of  column  distant  x  from  the  base,  for  column  hinged  at 
base ; 

MJ'  =  moment  at  any  section  of  column  distant  x  from  the  base,  for  column  fixed  at  base  ; 
MJ—  moment  in  column  at  foot  of  knee-brace  =  maximum  bending  moment  for  col- 
umn hinged  at  base ; 

MJ'=  moment  in  column  at  foot  of  knee-brace  —  maximum  bending  moment  for  col- 
umn fixed  at  base  ; 

M  =  the  sum  of  the  moments  of  the  horizontal  wind  loads  above  any  point  in  the  axis 
of  either  column  distant  x  above  the  base,  which  is  taken  as  the  centre  of 
moments  ;  note  especially  that  this  is  a  variable  quantity,  its  value  depending 
upon  the  height  of  the  point  taken  as  the  centre  of  moments  ; 

F,  =  the  vertical  reaction  at  either  column  due  to  the  overturning  action  of  the  wind 
on  one  entire  side  of  building  and  roof  for  columns  hinged  at  the  base  ; 
=  the  vertical  reaction  at  either  column  due  to  the  overturning  action  of  the  wind 
on  one  entire  side  of  building  and  roof  for  columns  fixed  at  base. 


IRON  AND  STEEL  MILL-BUILDING  CONSTRUCTION. 


FORMULAE. 

491.  Case  I.*— For  columns  hinged  at  the  base  and  considered  as  a  simple  beam  sup- 
ported at  both  ends: 

5;=   (0 

"      I  —  a 

P.^T^;  (2) 

I  —  a 

■      7',  =  ^^  

'  a 

M'x^a—S,X\   (3) 

M'^,,=^S,x-Pix-a);  (3^) 

M'^^^^S:{l-x);   {lb) 

M:  =  S,a\  (4) 

M:  =  S;{1  -a)  (4^^) 

m     hor.  comp.  of  wind  force  on  one  panel  of  building  and  roof  X  j  total  height 

V^  =  -l^  b  • 

This  vertical  reaction  is  constant  throughout  the  column  up  to  the  knee-brace.  Above 
that  point  it  is  increased  or  diminished  by  the  vertical  component  of  the  stress  in  the 
brace. 

492.  Case  II. t— For  columns  rigidly  fixed  at  the  base  and  considered  as  beams  fixed  at 
one  end  (the  base)  and  supported  at  the  other  (the  top) : 

—  PI 

=  — _      +      ;  o    .  (6) 


*  In  the  analysis  for  this  case  the  column  has  been  considered  as  a  beam  with  iinvicldinj;  supports;  strictly  this 
condition  will  not  be  realized,  for  as  the  top  of  the  column  deflects  to  the  leeward,  the  support  at  that  end  will  yield 
an  equal  amount,  and  the  resulting  stresses  will  be  somewhat  larger  than  found  by  (he  analysis.  However,  this 
increase  is  in  a  measure  counteracted  by  the  condition  of  partial  fixedness  at  the  top  of  the  column,  which  has  been 
disregarded  in  the  analysis. 

f  The  following  analysis  is  proximate.  It  assumes  that  the  top  of  the  building  does  not  deflect  laterally.  The 
rigid  analysis  for  the  general  case  is  found  in  Art.  412,  and  for  the  case  here  assumed  in  Art.  151. — J.  B.  J. 


47a 


MODERN  FRAMED  STRUCTURES. 


oX  taking  moments  about  the  base  of  column,  we  may  write,  in  terms  of      and  ^3', 

M,^  -  P,a-\-  S:i;  (6^) 

M" =  -  M,^S^',  (7) 

M\^.  =  -M,-^S,x-Plx-a);  

at  point  of  contraflexure  M^^  =z  —  M^-\-  S^x  =  o,  and 
M 

X  —       =1  distance  from  base  of  column  to  point  of  contraflexure  ;    .    ,    .    .  {yb'^ 

or  independently  in  terms  of  S^'  we  may  write 

^".>«  =S:{l-xy,  {yc) 

M\^.  =  S,V  -x)-P\a-x);  (7^ ) 

MJ'=-M,-^S,a;  (8) 

M:'  =  S,V-a);  (Sa) 

Solving  eq.  (9)  for  , 

p  —  ^   ;  (od) 

^-^[2k-3^'+n  +  P.k*  (10) 

The  vertical  reaction  is  best  found  for  the  point  of  contraflexure,  and  this  is  then  the 
vertical  stress  in  the  post,  from  wind,  from  the  base  to  the  foot  of  the  knee-brace.  While  the 
vertical  reaction  could  be  found  by  passing  a  horizontal  section  through  both  columns  at  any 
elevation,  and  summing  the  moments  of  the  external  forces  on  either  side  of  this  section,  we 
would,  in  general,  have  to  include  in  these  moments  the  bending  moments  at  this  section  in 
the  two  columns  cut.  Taking  the  section  through  the  points  of  contraflexure,  however, 
we  obtain 

V,=  ^at  point  of  contraflexure  or  when  x  =   (11) 

or,  so  far  as  the  overturning  action  of  the  wind  on  the  building  is  concerned,  the  columns  have 

M 

virtually  been  shortened  by  the  amount  x  =  ~,  which  is  the  distance  from  the  base  of 


*  Equations  (6),  (9),  and  (10)  are  expressions  for  a  beam  fixed  at  one  end  and  supported  at  the  other,  referred  to 
the  origin  of  co-ordinates  at  a  point  over  the  support  at  the  fixed  end,  and  can  be  found  in  text-books  on  the  subject, 
and  can  be  derived  from  the  formulx  given  in  Art.  131.  Valuable  tables  for  the  easy  solution  of  these  equations  are 
given  in  Howe's  "The  Continuous  Girder."  {Engbieering  News  Pub.  Co.,  1889.)  The  other  equations  under  Case  II 
and  all  the  equations  for  Case  I  can  be  written  directly  from  a  consideration  of  the  external  forces,  and  require  no 
extended  demonstration. 


IRON  AND  STEEL  MILL-B  UILDING  CONSTRUCTION. 


473 


column  to  the  point  of  contraflexure.  In  fact,  an  examination  of  the  several  equations  for 
moments,  shears,  and  horizontal  reactions  will  show  that  Case  II  becomes  in  all  respects 
Case  I,  with  the  base  of  column  moved  up  to  the  point  of  contraflexure. 

This  can  also  be  sliown  by  an  inspection  of  the  moment  and  shears  diagrams  on  Plate  I. 
This  is  true  not  only  for  the  moments  and  shears  for  the  part  of  the  column  above  the  point  of 
contraflexure,  but  also  for  the  stresses  in  trusses  and  knee-braces.  There  will  occur,  however, 
below  the  point  of  contraflexure,  shears,  and  a  negative  bending  moment,  of  which  the  action 
on  the  column  and  pier  must  be  considered. 

In  the  above  formulje,  several  expressions  have  been  given  for  the  values  of      and  V^. 

For  a  given  analysis,  however,  only  one  of  these  expressions  need  be  used  ;  the  position 
taken  for  the  centre  of  moments  determining  which  formula  shall  be  chosen. 

For  an  analysis  under  Case  I,  for  columns  hinged  at  the  base,  formulae  (i),  (2),  (4),  and 
(5)  only,  need  be  used;  and  for  an  analysis  under  Case  II,  formulae  (6),  (8),  {qo),  (10),  and 
(i  la)  only,  are  required.  The  supplementary  formulae  maybe  used  as  substitutes  for  those 
first  named  when  so  desired. 

The  horizontal  reaction  or  shear  at  the  base  of  a  column  is  a  known  quantity ;  it 
remains  the  same  whether  the  columns  are  hinged  or  rigidly  fixed  at  the  piers.  For 
the  two  columns  of  a  bent  it  is  equal  to  the  horizontal  component  of  the  wind  on  one 
panel  of  the  building  and  roof.  This  total  shear  or  reaction  is  assumed  to  be  equally 
distributed  between  the  two  columns.  The  wind  loads  are  considered  concentrated,  and  the 
concentration  at  the  foot  of  the  column  is  disregarded  in  the  analysis  of  stresses  in  trusses 
and  columns,  but  is  considered  in  the  calculation  for  anchor-bolts,  masonry,  etc. 

493.  External  Forces. — The  columns  in  deflecting  from  the  wind-loads  have  a  leverage 
action  producing  certain  horizontal  reactions  at  the  foot  of  the  knee-braces  and  at  bottom 
chord  of  roof  truss,  which  must  be  considered  as  external  forces,  in  finding  the  vertical  reac- 
tions and  in  the  analysis  for  stresses  in  the  knee-braces  and  roof  truss. 

494.  Vertical  Reactions  from  Horizontal  Forces. — Taking  the  centre  of  moments  at 
any  point  in  the  axis  of  either  column  the  vertical  reaction  at  the  opposite  column,  in  any 
case  is  equal  to  the  algebraic  sum  of  the  moments  of  all  the  external  forces  above  or  below 
this  point  plus  the  sum  of  the  moments  in  the  two  columns  themselves  at  this  horizontal  plane, 
divided  by  the  span  of  the  building.  The  centre  of  moments  may  be  taken  at  any  point  in 
the  axis  of  the  column,  but  the  most  convenient  points  are  at  the  base  of  the  column,  at  the 
point  of  contraflexure,  and  at  the  foot  of  the  knee-brace.  When  taken  at  this  point  of  contra- 
flexure the  moments  in  the  columns  themselves  disappear. 

495.  Vertical  Reactions  from  Vertical  Forces. — The  vertical  reaction  at  either 
column  resulting  from  the  vertical  components  of  the  horizontal  wind-force  is  equal  to  the 
sum  of  the  moments  of  these  vertical  forces  about  a  point  in  the  axis  of  the  other  column, 
divided  by  the  span  of  the  building. 

For  analysis  in  which  only  the  horizontal  component  of  the  wind-force  is  dealt  with,  as 
in  Cases  I  and  II,  the  vertical  reactions  resulting  from  the  horizontal  component  only  need 
be  considered.  When,  however,  as  in  Case  V,  the  analysis  is  made  for  the  horizontal  and 
vertical  components  of  the  resultant  normal  wind  pressure,  the  algebraic  sum  of  the  corre- 
sponding vertical  reactions  must  be  considered.  In  all  cases  the  wind  and  dead  load  stresses 
have  been  determined  separately  for  the  purpose  of  comparison  and  to  locate  the  alternating 
stresses.  To  determine  the  net  stresses  in  the  members  and  the  resultant  overturning  action 
on  the  building,  the  stresses  and  reactions  for  dead  and  wind  loads  must  be  combined 
algebraically. 

496.  Analytical  Process  for  Case  I. — Find  5, ,  the  shear  at  the  base  of  the  column,  and 
substitute  its  value  in  equations  (i),  (2),  and  (4).  Solve  (1)  for  5/,  the  shear  at  the  top  of  the 
column.  Solve  (2)  for      ,  the  horizontal  load  at  the  foot  of  the  knee  brace.  Solve  (4)  for  M\, 


474 


MODERN  FRAMED  STRUCTURES. 


the  maximum  positive  bending  moment  in  the  column.  This  bending  moment  occurs  at  the 
foot  of  the  knee-brace. 

Find  w,  the  sum  of  the  moments  of  the  horizontal  wind  loads,  taking  the  foot  of  one  of 
the  columns  as  the  centre  of  moments  (see  nomenclature),  and  divide  by  b,  the  span  of  the 
building  ;  this  gives  F, ,  the  vertical  reaction  at  the  foot  of  either  column  due  to  the  horizontal 
component  of  the  wind  force,  as  shown  in  equation  (5). 

497.  Analytical  Process  for  Case  II. — Find  the  shear  at  the  base  of  the  column, 
and  substitute  its  value  in  equation  {<^d),  and  solve  for  the  load  at  the  foot  of  the  knee- 
brace. 

Substitute  value  of  in  (6)  and  (10),  and  solve  (6)  for  ,  the  negative  bending  moment 
at  the  foot  of  the  column  ;  and  solve  (10)  for  S^,  the  shear  at  the  top  of  the  column.  Substi- 
tute values  of  and  in  (8),  and  solve  for  M^' ,  the  maximum  positive  bending  moment  in 
the  column.  This  bending  moment  occurs  at  the  foot  of  the  knee-brace.  Finally  find  the 
sum  of  the  moments  of  the  horizontal  wind  loads  (see  nomenclature),  and  substitute  the 
values  of  m  and  in  {\\b\  and  solve  for  F, ,the  vertical  reaction  at  foot  of  either  column  due 
to  the  horizontal  component  of  the  wind  force.  (See  paragraph  relating  to  "  Vertical  Reac- 
tions from  Vertical  Forces.") 

498.  Graphical  Process  for  Cases  I  and  II. — The  action  of  the  horizontal  forces  on 
the  columns  has  now  been  fully  determined.  The  horizontal  and  vertical  reactions  have  also 
been  determined,  and  the  stresses  in  roof  trusses  and  knee-braces  can  be  easily  found  graph- 
ically. The  graphical  treatment  is  shown  in  full  on  Plate  XXXII,  Fig.  2.  Taking  the  foot  of 
the  windward  knee-brace  as  a  convenient  starting-point,  the  force  polygon  is  constructed  as 
follows :  From  a  X.o  b  \.o  c-d-e-f-x-y-t  to  a  to  close  at  the  point  of  starting,  the  polygon  being 
made  up  of  the  wind  loads  and  the  horizontal  and  vertical  reactions  mentioned  above. 

The  shear  at  the  base  of  the  columns  does  not  directly  appear  in  the  force  polygon,  or  in 
the  stress  diagram,  but  its  equivalent  has  been  considered  in  the  load  at  the  foot  of  the  knee- 
brace  and  in  the  shear  at  the  top  of  the  columns. 

Notice  that,  at  the  foot  of  the  windward  knee-brace,  the  concentrated  wind  load  at  that 
point  must  be  deducted  from  the  load  or  P^  to  find  the  net  horizontal  reaction  which  is  to 
be  used  in  the  force  polygon,  and  which  is  the  horizontal  component  of  the  stress  in  the  wind- 
ward knee-brace. 


1 

_  r^/af^  32 


t 

/  \ 

1 

^     ^ stresses,  CaseS 


'frdss  Mifml>tr  Sfuss. 


^7300  O  B  ■i-S4&nO 


zsae 


y-z 


i32ffff\  ii/-a  +3s3eff 


?AWi  Jity^  -2220P 


Z3600  [  J/-JL_  +24000 


V  5    +  2sao 


AAA  A 

^  J,  


STRUCTURAL  STEEL 


AI^D  GENERAL  SPECIFICATIONS, 


APPENDIX  A. 


STRUCTURAL  STEEL  AND  GENERAL  SPECIFICATIONS* 

SOFT  STEEL  IN  BRIDGES.! 

It  is  not  to  be  expected  that  a  bridge  specification,  prepared  at  this  time,  will  contain  much  that  is  new 
or  novel.  It  will,  no  doubt,  contain  something  of  the  individual  tastes  and  preferences  of  the  party  who  ex- 
pects to  use  it.  If  the  writer  is  fortunate  enough  to  include  the  approved  features  of  existing  specifications, 
a  new  idea  or  two  will  quite  suffice  for  novelty.  The  accompanying  specification  is,  in  fact,  a  revision  of 
an  older  one,  which  in  turn  derived  its  salient  features  from  the  well-known  specification  of  the  Keystone 
Bridge  Company.  It  is  not  the  writer's  purpose,  therefore,  to  enlarge  upon  the  subject-matter  in  any 
general  or  extended  way,  but  to  confine  his  remarks  chiefly  to  one  feature.  It  will  be  evident  to  any  one 
who  examines  the  unit  stresses  that  they  are  expressly  intended  to  put  soft  steel  on  at  least  an  equal  footing 
with  wrought-iron  and  medium  steel ,  and  the  idea  in  laying  the  specifications  before  the  Club  at  this  time 
was  to  have  this  feature  fully  discussed  for  mutual  benefit.  While,  therefore,  any  part  of  the  text  is  open 
for  criticism  or  suggestion,  the  steel  question  is  the  primary  one  ;  and  to  that  alone  the  writer  proposes  to 
address  himself. 

There  are,  at  this  time,  three  grades  of  material  offered  to  engineers  for  structural  work — viz.,  wrought- 
iron,  soft  steel,  and  medium  steel.  Of  these  we  are  using  wrought-iron  for  short  spans  and  medium  steel 
for  long  spans,  while  the  soft  steel,  which  is  in  many  respects  the  best  grade  of  material  of  the  three,  we  are 
using  very  little  r  and  there  seems  to  be  no  immediate  prospect,  under  existing  specifications,  of  extending 
its  use.  It  has  oeen  the  writer's  good  fortune  to  have  an  intimate  acquaintance,  at  first  hand,  with  the  charac- 
ter of  the  material  now  being  turned  out  by  our  manufacturers,  and  to  have  been  much  impressed,  in  conse- 
quence, with  the  idea  that  in  neglecting  to  use  soft  steel  we  are  doing  so  to  the  disadvantage  of  our  bridge 
structures,  and  that  it  is  really  important  to  give  it  a  standing  in  specifications  which  will  allow  it  to  be 
used  on  at  least  equal  terms  with  the  other  grades  of  material.  Our  wrought-iron  is  certainly  not  improv- 
ing in  quality,  and  in  certain  directions  it  is  distinctively  poorer  than  it  formerly  was.  In  the  manufacture 
of  steel,  however,  there  has  been  a  notable  improvement  in  quality  and  uniformity,  and  the  best  qaalities  of 
soft  steel  now  leave  very  little  further  to  be  desired.  As  compared  with  wrought-iron,  we  have  a  material 
which,  with  15  percent  higher  ultimate  strength,  has  from  20  to  25  per  cent  higher  elastic  limit,  and  from 
50  to  60  per  cent  greater  ductility.  We  have  a  material  which  has  equal  strength,  ductility,  and  bending 
qualities,  lengthwise,  and  crosswise  of  the  material,  in  comparison  with  wrought-iron,  which  has  little 
strength  or  ductility  and  will  not  bend  across  its  grain.  We  have,  again,  a  material  which  has  no  fibre,  and 
which,  consequently,  has  less  disposition  to  tear  or  pull  out,  and,  lastly,  we  have  the  fine  finish  and  sound, 
clean  metal  characteristic  of  steel,  in  comparison  with  the  rougher  finish  and  frequently  doubtful  welding  of 
wrought-iron. 

^  Now,  in  certain  well-known  specifications,  and  with  the  tacit  consent  of  many  engineers,  our  bridge 
builders  have  been  privileged,  for  several  years  past,  to  use  soft  steel  on  the  same  basis  as  wrought-iron.  At 
no  time,  however,  have  they  taken  advantage  of  this  privilege  to  any  great  extent,  and  it  seems  quite  uncer- 
tain whetiier  they  will  do  so.  We  get  a  little  soft  steel  occasionally  when  there  are  long  bars  to  be  rolled, 
which  can  be  gotten  out  easier  in  steel,  or  when  there  is  a  large  order  of  plates  that  can  be  rolled  more  eco- 
nomically, and  we  get  occasionally  pieces  of  steel  in  our  structures  for  other  purposes  ;  but  anything  like  a 
general  use  of  it  has  not  come  to  pass,  and  the  very  simple  reason  is  that  it  costs  more.  The  writer  has 
endeavored  to  learn  from  our  manufacturers  whether  the  relative  difference  between  wrought-iron  and  soft 
steel  would  be  likely  to  decrease  soon,  but,  so  far  as  he  has  been  able  to  ascertain,  none  of  them  are  prepared. 


*  This  appendix  is  adapted  from  a  paper  which  was  read  before  the  Engineers'  Club  of  Philadelphia,  by  M.  F.  H. 
Lrwis,  October  17,  1891.  See  also  paper  by  Mr.  Lewis  before  Engrs.  Soc.  W.  Penn.,  in  Engineering  News,  April  25, 
•b95  (Vol.  XXXIU,  p.  276). 

•f- The  reader  i-<  r'  ferr'd  10  Pn.f.  Ii.h'  son's  /lAz/crzaA'  0/ Cf»f/r«f/«<»w  (1897)  for  a  full  working  knowledge  of  ail 
kinds  <i(  sUc.  .iiid  oilier  Uiiid^  of  Iniililing  m.ilerial. 


4?6 


APPENDIX. 


to  commit  themselves  on  this  subject.  It  seems  liicely,  if  there  is  a  considerable  market  for  it,  under  stand, 
ard  specifications,  that  the  cost  will  be  reduced  to  that  of  wrought-iron,  or  lower;  but  it  also  seems  liicely 
that  this  will  not  transpire  until  we  do  use  it  and  cheapen  it  by  making  a  market  for  it.  Within  the  last 
few  years  quite  a  number  of  engineers  have  evidently  formed  favorable  opinions  of  this  grade  of  material 
and  have  essayed  to  use  it,  but  under  conditions  which  do  not  seem  to  be  entirely  warranted.  In  one  or 
two  cases  they  are  reaming  it,  just  as  they  would  medium  steel;  but  this  can  hardly  be  regarded  as  an 
advantageous  procedure,  because,  first,  the  metal  is  not  of  a  character  adapted  to  reaming,  being  of  a  soft, 
waxy  texture,  which  drags  on  a  tool  and  clogs  it,  making  more  expensive  work  than  medium  steel ;  and, 
second,  we  are  not  warranted  in  figuring  it  at  high  enough  unit  stresses  to  cover  this  expense.  We 
might,  perhaps,  rate  the  unit  stresses  at  15  per  cent  higher  than  wrought-iron  ;  but  such  a  percentage  will 
certainly  not  pay  the  cost  of  reaming,  added  to  the  extra  cost  of  the  material.  Again,  several  engineers 
have  been  using  soft  steel  under  the  same  conditions  as  wrought-iron,  but  with  the  higher  stresses  wliich 
have  been  adopted  for  medium  steel.  It  is  an  open  question  whether  such  a  course  as  this  is  warranted  by 
the  facts  of  the  case.  It  is  not  necessary,  in  the  first  place,  to  figure  it  so  high  to  place  it  on  a  par  in  cost 
with  wrought-iron;  and  then  the  material  is  certainly  not  of  equal  value  with  reamed  medium  steel,  for  the 
reasons,  first,  that  its  elastic  limit  and  ultimate  strength  are  both  lower,  and,  second,  because  reaming 
undoubtedly  adds  to  the  value  of  any  grade  of  steel,  or  of  wrought-iron,  for  that  matter. 

The  correct  solution  of  the  matter  would  appear  to  be  to  strike  a  balance  between  the  cost  of  soft  steel  in 
bridge  structures,  as  compared  with  wrought-iron  on  the  one  hand,  and  medimn  steel  on  the  other,  and  to  rate 
it  accordingly  if  we  are  fully  warranted  to  do  so,  at  unit  stresses  high  enough  to  overcome  the  difference  in 
cost.   The  present  difference  in  cost  between  wrought-iron  and  soft  steel  may  be  reckoned  about  as  follows : 

Cost  of  wrought-iron,  say   100 

Increased  weight  of  steel  2  per  cent     2 

Add  increased  cost  of  soft  steel  5  per  cent  (not  well  maintained)   5.1 

Total  ,.   107. 1 


In  other  words,  the  cost  of  soft  steel  is  nominally  a  little  over  7  per  cent  higher  than  wrought-iron.  In 
the  shop  there  would  be  a  slight  saving  in  using  soft  steel  due  to  a  rivet  saved  here  and  there,  and  other 
small  savings,  provided  we  can  use  it  just  like  wrought-iron.  The  writer  takes  it  for  granted  that  the  pro- 
priety of  using  soft  steel,  havingan  average  ultimate  strength  of  58,000  pounds,  at  unit  stresses  in  compression 
10  to  15  per  cent  higher  than  wrought-iron,  will  not  be  seriously  questioned  ;  that  is  to  say,  under  the  actual 

/ 

conditions  in  which  it  is  used  in  our  bridges,  or  with  values  of  —  not  exceeding  150.    The  propriety  of  using 

r 

soft  steev  under  higher  unit  stresses  than  wrought-iron  hinges  then  upon  its  qualities  in  tension  in  flanges 
of  girders  and  in  the  tension  members  of  trusses  ;  and  this  phase  of  the  subject  will  now  be  discussed  some- 
what at  length : 

Whether  Soft  Steel,  after  Punching,  constitutes  a  Good  Tension  Member. 

If  left  to  our  practical  shopmen,  there  would  probably  be  no  hesitation  whatever  on  this  score.  It  is, 
perhaps,  no  exaggeration  to  say  that  for  exacting  conditions  of  all  sorts  it  is  becoming  more  and  more  the 
practice  to  substitute  soft  steel.  It  is  being  used  for  latticing,  because  it  will  not  split ;  it  is  being  used  for 
the  hitch  angles  of  stringers  and  floor-beams,  because  it  will  not  crack  ;  it  is  being  used  for  pin  plates  o'n 
suspenders,  and  in  various  other  ways  where  its  superior  qualities  have  been  practically  shown  to  practical 
men.  It  is  necessary  to  get  more  conclusive  evidence  than  this,  however,  and  the  presentation  of  this  evi- 
ience,  as  I  have  been  able  to  collect  it,  follows  below.  The  writer  wishes  it  understood  at  the  outset  that 
in  nothing  which  follows  is  there  any  attempt  to  show  that  reaming  is  not  a  good  thing  per  se.  On  the 
contrary,  the  tests  given  below  show  that  it  generally  benefits  soft  steel  whenever  the  metal  is  over  thick. 
It  is  well  to  state  this  very  clearly,  because  otherwise  the  discussion  might  hinge  on  the  relative  merits  of 
punching  and  reaming,  which  is  not  now  at  issue.    We  cannot  ream  soft  steel  and  use  it  economically. 

The  question  is  not,  therefore,  whether  punching  is  better  than  reaming,  but  whether  we  are  as  fully 
warranted  to  punch  soft  steel  as  we  are  to  punch  wrought-iron.  We  are  punching  iron  for  all  conditions  of 
work  ;  and  if  soft  steel  holds  its  value  after  punching  as  well  as  or  better  than  iron,  then  we  are  clearly  free  to 
use  it  in  the  same  way.    To  test  the  question  it  is  only  necessary  to  take  tested  samples  of  iron  and  steel, 


STRUCTURAL  STEEL  AND  GENERAL  SPECIFICATIONS. 


All 


and  after  punching  them  under  the  same  conditions,  again  test  them,  and  by  simple  rule  of  three  ascertain 
which  holds  its  value  best.  This  has  been  done  in  the  tables  which  follow.  Except  where  noted  in  the 
tables,  the  values  obtained  in  specimen  tests  are  always  rated  at  loo,  and  the  values  after  punching  are  rated 
by  percentage  from  this  basis. 

Table  I. 

U.  S.  GOVERNMENT  TESTS  OF  RIVETED  LAP-JOINTS  AT  WATERTOWN  ARSENAL 

Reports  of  1882-83. 


PUNCHED  HOLES. 


Grooved  Iron. 

Grooved  Steel. 

Specimen. 

Kind  of  Joint. 

Specimen  Test,  j 

Net  Section 
of  Joint. 

Percentage 

Specimen  Test,  j 

Net  Section 
of  Joint. 

Percentage. 

Ultimate. 

Ultimate. 

10"  X  J"  plate  \ 
I 

\ 

Single  lap 
Double  lap  | 

47.925 
1* 

43.230 
45,520 
52, 160 
54.930 
50,592 

Mean  = 

90 

95 
109 

115 
106 
104 

103 

55.765 

60,2^0* 
59.240 

61,510 

60,300 

Mean  = 

106 

110 

108 

1 08 

10"  X  j"  plate  1^ 

I 

Single  lap 
Double  lap 

47,180 

36,130 
41.750 
41,290 
61,200 
58.510 
48,450 
50,730 
46.255 
46, 1 10 

Mean  ~ 

77 
89 
87 
131 
124 
103 
108 
98 
98 

102 

53.330 

(< 
,1 

55,215* 

65,460 

65,210 

73,394 

73.970 

62,800 

64,720 

63,210 

54,930 

Mean  = 

123 
122 
138 
139 
118 
121 
119 
103 

121 

10"  X  \"  plate  1 

Single  lap  | 

r 

Double  lap 

44,615 

3 1 , 1 00 
3', 395 
35,650 
44.320 
42,920 
46,400 
46, 140 

Mean  = 

70 
70 
80 

99 
96 
104 
104 

89 

57.215 

<< 

60,210 

49.590 

47.530* 

64,602 

64,519 

68,920 

66,710 

Mean  = 

105 

87 

113 
113 
120 
118 

109 

10"  X  i"  plate  1 

Single  lap  | 
Double  lap  | 

44.630 

34,680 
34,230 
43.580 
45.850 

Mean  = 

78 
77 
98 
103 

89 

52.445 

52,770* 
49,610* 
69,680 
67, 100 

Mean  = 

133 
128 

131 

10" X  f "  plate  1 

Single  lap  j 
Double  lap  | 

46,590 

2g,2go 
30,730 
42,000 
43.950 

Mean  = 

63 
66 

90 
94 

78 

51-545 

39.970 
47.370 
48,970 
47.510 

Mean  = 

78 
92 
95 
92 

89 

*  Did  not  rupture  the  plate. 


This  table  is  derived  from  a  series  of  tests  of  riveted  joints  which  appear  in  the  Government  Reports 
of  tests  made  at  the  Watertown  Arsenal.  In  this  series  of  tests,  with  which  many  members  are,  no  doubt, 
familiar,  steel  and  wrought-iron  were  compared  under  practically  the  same  conditions  through  a  long  series 


478 


APPENDIX, 


of  tests  of  both  punched  and  reamed  material,  and  the  ultimate  strengths  of  the  samples  were  very  carefully 
determined  by  tests  on  grooved  specimens.  The  conditions  were  not  always  exactly  identical,  as  the  rivets 
were  sometimes  of  wrought-iron  and  sometimes  of  steel,  and  the  joints  failed  accordingly,  sometimes  by 
shearing  the  rivets  and  sometimes  by  tearing  the  plate.  I  have  endeavored,  however,  in  this  table  to  com- 
pare, on  essentially  similar  terms  in  each  case,  the  behavior  of  the  wrought-iron  and  of  the  steel.  The 
table  includes  every  punched  test  in  the  series  which  failed  by  tearing  across  the  sheet,  and  also  includes  a 
lew  tests  which  failed  by  shearing,  but  which  probably  developed  nearly  the  full  value  of  the  material. 
These  last  tests  are  indicated  by  stars.  Unfortunately,  the  elastic  limit  was  not  determined  in  these  tests, 
and  no  comparison  can  therefore  be  made  between  the  elastic  limit  of  the  specimens  and  the  elastic  limit  of 
the  riveted  joints.  The  uitimates,  therefore,  only  are  compared,  and  the  relative  ultimate  strength 
developed  in  net  sections  at  the  joints  is  figured  out  in  the  columns  headed  "  Percentage"  throughout  the 
table.  It  will  be  found  that  not  only  does  the  steel  show  less  loss  in  ultimate  strength  as  an  effect  of 
punching  than  the  wrought-iron,  from  an  average  of  these  tests,  but  in  every  identical  test,  with  one 
exception,  this  fact  is  also  true,  and  as  the  thickness  of  the  pieces  ranges  froms  J"  up  to  f",  the  series  is  very 
complete.  It  is  true  some  of  this  steel  is  a  little  softer  than  the  soft  steel  which  it  is  proposed  to  use  for 
bridge  structures ;  but,  on  the  other  hand,  the  iron  is  also  of  good,  soft  quality,  probably  boiler-plate  iron, 
as  the  tests  indicate  it  to  have  an  unusually  large  percentage  of  strength  across  the  grain.  It  would, 
therefore,  probably  give  better  results  than  ordinary  wrought-iron  would,  and  the  comparison  is  therefore 
a  fair  one. 

[Note. — In  Table  I«are  given  the  results  of  tests  on  iron  and  steel  single-riveted  butt-joints  with  punched 
holes,  as  compared  with  the  strength  of  the  grooved  plates.  This  is  the  kind  of  riveted  joints  used  in 
structures,  and  hence  indicates  more  fully  the  relative  effect  of  punching  iron  and  steel  in  structural  work 
than  the  tests  in  lap-joints  given  in  Table  I. — J.  B.  J.J 


Table  \a. 


RELATIVE  STRENGTH  OF  IRON  AND  STEEL  SINGLE-RIVETED  BUTT-JOINTS  WITH 

PUNCHED  HOLES.* 


Iron  Plates. 

Steel  Plates. 

Thick- 
ness of 
Plate. 

Diameter 
of  Rivet- 
holes. 

Pitch 

of 
Rivets. 

No.  of 
Tests. 

Strength  of 
Grooved 
Plate. 

Strength  of 
Net  Section 
of  Riveted 
Joint. 

Excess 
Strength  of 
Joint. 

Per  cent 
of  Excess. 

Strength  of 
Grooved 
Plate. 

Strength  of 
Net  Section 
of  Riveted 
Joint. 

Excess 
Strength  of 
Joint. 

Per  cent 
of  Excess 

1  in. 

f  in. 

2  in. 

2 

Pounds 
per  sq.  in. 

47,i8o 

Pounds 
per  sq.  in. 

46,620 

Pounds 
per  sq.  in. 

—  560 

—  1.2 

53,330 

62,000 

+  8,670 

+  16.2 

\  in. 

1  3  in 

2  in. 

2 

44.615 

46,270 

+  1,655 

+  3.6 

57,215 

67,820 

-f-  10,605 

+  18.5 

i  in. 

IjV  in- 

2\  in. 

2 

44,635 

43,300 

-  1.335 

-  3-1 

52.445 

62,380 

+  9,835 

-f  18.8 

fin. 

Itf  in- 

2f  in. 

2 

46,590 

42,020 

-  4,570 

—  10.9 

51.545 

54-425 

+  2.875 

+  5-6 

[Tliis  table  shows  that  whereas  wrought-iron  riveted  plates  are  weaker  than  the  grooved  specimen  tests, 
ihe  corresponding  steel-riveted  plates  are  much  stronger. — J.  B.  J.] 

Table  II  is  also  from  the  Government  tests  at  Watertown,  and  consists  of  comparative  results  obtained 
from  grooved  specimens  alternately  drilled  and  punched.  The  percentages  of  this  table  are  made  up  by 
rating  the  uitimates  of  the  drilled  specimens  at  100,  and  in  this  table  also  it  will  be  found  that  the  steel  is 
less  impaired  by  punching  than  the  wrought-iron.  The  steel  and  iron  used  in  these  tests  are  the  same 
material  as  that  tested  in  Table  I. 


*  From  Watertown  Arsenal  Reports  of  Tests  of  Metals,  etc.,  for  1882  and  1883,  given  also  in  Lanza's  "Mechanics." 


STRUCTURAL  STEEL  AND  GENERAL  SPECIFICATLONS 


Table  II. 


U.  S.  GOVERNiMENT  TESTS  OF  GROOVED  SPECIMENS  AT  WATERTOWN  ARSENAL 

Report  of  1802. 


Shape  of  specimen  : 


1  to  4 

_A_ 


Drilled. 

44,980 
47.030 
45.330  ■ 
45,000 
46, 1 00  ' 


47,220 
48,350 
47,170 
46,350 
48,220 
44,840  I 

45.100  s 


47.500 

52,780 
48,470 
47.750 
46.350 


Iron. 

Punched. 

38,950 

37,200 

35.730] 

36,690  ' 

37,000 

37.420 


49.770 
52,960 
46,320 

46,750 
40,140 
37.310 


50,840 
47.590 
45.970 
40,350 
39.380 


Percentage. 

86 
79 


82 


105 
1 10 
98 

lOI 

83 
83 


97 


107 
88 
95 
84 
85 

92 


Specimen, 
i"  X  3"  wide 
fx 4"  " 


I"  X  i''  wide 


I"  X  3" 
f '  X  4" 


f"  X  i"  wide 

J"  X  li"  •• 

5"  X  3i  '  " 


Drilled. 

66,310 
66, 190 
64.470 
64,810 
64,690 
64, 140 


60, 290 
63,610 
63.450 

59,270 
60,330 
61, 120 


58,480 
58,790 
59.290 
58,700 
59,180 


Steel. 
Punched. 

60,320 
62,430 
48,010! 
55.190  I 
55,780  I 
46, 250  J 


66, 720 
64,800 
64,400 

57,180 
54.450 
57.380 


67,930 
67,630 
62,890 
56,730 
54,220 


[Note.— Table  II  might  liave  been  greatly  extended  by  including  the  tests  reported  in  1883.  The 
following  table  gives  the  summaries  by  averages  of  all  such  tests  made  at  the  Watertown  Arsenal. — J.  W.  J.] 

Table  \\a. 

RESULTS  OF  TESTS*  SHOWING  EFFECTS  OF  PUNCHING  VVROUGHT-IRON  AND  MILD-STEEL 

PLATES. 


Summary  of  Watertown  Arsenal  Tests. 
Shape  of  specimen  : 


l"to  4" 
_f*L_ 


Thick- 
ness of 
Plate. 

Width  at  Bottom 
of  Groove. 

Iron  Plates. 

Steel  Plates. 

No.  of 
Tests. 

Average 
Strength 
Drilled. 

Average 
Strength 
Punched. 

Loss  from 
Punching. 

Percent- 
age 
of  Loss. 

No.  of 
Tests. 

Average 
Strength 
Drilled. 

A  vcr.Tge 
Strength 
Punched. 

Loss 
from 
Punch- 
ing. 

Percent- 
age 
of  Loss. 

Pounds 

Pounds 

Pounds 

Pounds 

Pounds 

Pounds 

per  sq  in. 

per  sq  in. 

per  sq.  in. 

per  sq.  in. 

per  sq.  in. 

per  sq. in. 

\  in. 

\  in.  to  3  in. 

42 

50, 100 

43.500 

6,600 

13.2 

31 

64,200 

61,800 

2,400 

3-7 

1  in. 

I  in.  to  3^  in. 

9 

48,500 

40,000 

8,500 

175 

I 

63,620 

61,890 

1.730 

2.7 

A  in. 

I  in.  to  4  in. 

15 

47,100 

40,300 

6,800 

14-5 

17 

66,030 

58,160 

7.870 

12.0 

1  in. 

I  in.  to  4  in. 

8 

46,670 

43.460 

3,210 

6.9 

10 

61,050 

59,740 

1,310 

2.1 

(Gain) 

Gain  of 

f  in. 

I  in  to  3i  in. 

5 

48,570 

44.650 

3,920 

8.1 

5 

58,890 

61,880 

2,990 

51 

*  From  Watertown  Reports  of  Tests  of  Metals,  etc.,  for  1882  and  1883  ;  given  also  in  Lanz.n's  "  Mechanics." 


480 


STRUCTURAL  STEEL  AND  GENERAL  SPECIFICATIONS. 


[Note. — By  comparing  Tables  \\a  and  \a  we  see  that  wrought-iron  punched  plate  is,  say,  12  per  cent 
weaker  than  the  same  drilled,  and  that  the  riveted  sheet  is,  say,  4  per  cent  weaker  than  the  punched  plate, 
or  the  total  weakening  of  wrought-iron  has  been  some  16  per  cent,  while  there  has  been  a  corresponding  loss 
m  strength  of  the  punched  steel  plates  of,  say,  3  per  cent,  but  digam  in  strength  of  the  riveted  steel  plate 
over  the  punched  sheet  of,  say,  15  per  cent,  or  a  net  gain  of  so)?te  12  per  cent.  This  is  the  average  gain  in 
strength  of  a  punched  and  riveted  steel  plate  as  compared  with  the  drilled  grooved  specimen. — J.  B.  J.] 

In  these  Government  Reports  will  also  be  found  some  interesting  tests  on  the  tearing  out  of  rivet-holes 
n  steel  and  iron.  Unfortunately  the  series  is  limited  to  J"  metal,  and  is  consequently  of  limited  value, 
in  J"  metal  the  difference  between  punched  and  drilled  holes  is  very  small  indeed.  So  far  as  these  tests  go, 
however,  they  show  that  steel  with  punched  holes  and  sheared  edges  gives  better  relative  results  than  iron  with 
drilled  holes  and  planed  edges. 

Not  long  since,  it  was  the  writer's  good  fortune  to  find  a  very  interesting  series  of  comparative  tests 
of  iron  and  steel  on  record  at  one  of  the  larger  iron-works  of  this  district,  and  through  the  courtesy  of  the 
proprietors  he  has  been  able  to  examine  it  to  determine  the  same  ratios  given  in  the  tables  above.  As 
the  complete  results  will  probably  be  published,  the  details  will  not  here  be  given,  but  only  the  general 
results.  The  tests  are  important,  because  punched,  reamed,  and  drilled  specimens  of  iron  and  steel  are  com- 
pared under  exactly  similar  conditions,  and  the  series  comprises  some  fifty  or  sixty  tests.  The  bars  used  for 
the  purpose  were  6  x  |,  6  x  f,  and  6  x  f  respectively  in  both  iron  and  steel,  the  steel  having  an  ultimate 
strength  of  about  64,000  pounds,  and  the  iron  an  ultimate  strength  of  about  52,000  pounds,  and  both  grades 
of  material  were  of  excellent  quality.  The  writer  has  checked  up  the  entire  series  and  finds  that  the  steel  is 
very  uniformly  less  injured  by  punchitig  than  iron  ;  the  average  difference  being  about  lo  per  cent  in  favor  of 
steel,  and  this  difference  is  essentially  true  of  all  tests  save,  perhaps,  one  in  the  series.  Thus,  if  we  rate  the 
average  ultimate  strength  of  the  steel  in  specimen  tests  at  100,  the  average  ultimate  of  the  iron  in  specimen 
tests  would  rate  at  83 ;  but  rating  the  average  ultimate  of  the  punched  steel  on  net  sections  at  100  per  cent, 
the  average  ultimate  of  punched  iron  on  net  sections  would  rate  at  per  cent.  More  than  this,  a  com- 
parison of  the  punched  specimens  with  the  reamed  ones  in  both  cases  shows  the  iron  to  have  been  quite  as 
much  benefited  by  reaming  as  the  steel. 

In  Table  III  are  given  some  tests  of  wrought-iron,  soft  steel,  and  medium  steel,  which  have  been  made 
recently  under  the  auspices  of  the  Pittsburgh  Testing  Laboratory.  Some  of  the  specimens  were  tested  with 
open  holes  and  some  with  rivets  in  the  holes.  In  these  tests  the  elastic  limit  was  taken  in  all  cases,  and  it 
will  be  observed  that  with  punched  holes  the  material  retains  its  full  value  at  the  elastic  limit,  and  indeed 
generally  records  a  gain  in  this  respect,  showing  the  punching  to  have  made  the  metal  harder  and  denser. 
In  the  series  of  tests  with  rivets  driven  in  the  holes,  it  is  evident  that  generally  the  rivet  acted  as  a  splice, 
and  the  elastic  limit  is  really  that  of  the  gross  section.  The  tests  on  their  face  hardly  indicate  as  high 
values  as  this,  but  the  fact  seemed  to  be  as  stated.  The  heating  in  riveting  no  doubt  softened  the  metal  and 
reduced  the  elastic  limit. 

In  Table  IV  a  number  of  tests  of  punched  medium  steel  are  given,  and  also  two  of  hard  steel.  With  the 
exception  of  the  latter  grade  of  metal,  a  comparison  of  results  decidedly  favors  the  steel. 

It  will  be  evident  to  any  one  who  examines  the  facts  here  presented  that  there  are  several  points 
suggested  by  these  tests  which  are  not  proved,  and  which  can  only  be  demonstrated  by  more  extended  tests. 
The  field  for  such  experiments  is  evidently  an  ample  one.  But  on  the  definite  question  raised  in  this  paper 
may  it  not  be  said  that  the  tests  presented,  whether  considered  as  a  whole  or  in  detail,  SlVC  positive  evidence 
that  the  tensile  strength  of  structural  steel  is  injured  rather  less  by  punching  than  is  the  tensile  strength  of 
wrought-iron?  It  is  not  the  writer's  intention  to  claim  that  this  demonstration  is  conclusive,  much  less 
exclusive  of  testimony  to  the  contrary.  But  it  is  entirely  true  so  far  as  the  evidence  goes  ;  and  this  evidence, 
as  presented,  is  entirely  full  and  candid,  including  everything  that  the  writer  was  able  to  find  bearing  on  this 
subject,  and  the  entire  series  of  ninety  special  tests  made  under  his  own  direction.  He  has  no  idea  that  it  can 
be  successfully  gainsaid.    As  conclusions  suggested,  but  not  demonstrated,  he  would  state  the  following: 

(1)  The  thicker  the  inetal  the  greater  the  injury  in  punching  steel. 

(2)  The  softer  the  steel  the  less  the  itijury  in  punching  ;  but, 

(3)  The  first  conclusion  is  the  tnajor  factor  ;  the  grade  of  the  steel  making  viuch  less  difference  than  the 
thickness  of  the  material. 

(4)  The  effect  of  riveting  is  advantageous,  increasing  the  strength  of  net  sections. 

Conclusions  i  and  2  have  ample  support  from  other  sources  and  may  be  considered  as  facts.  Thus, 
Kirkaldy's  recent  book,  in  commenting  on  his  well-known  series  of  tests  of  Fagersta  plates,  speaks  as 
follows:  "  Punching  is  less  detrimental  the  softer  the  plate;  the  great  difference  produced  by  punching  is 
due  to  its  hardening  effect  upon  the  plates,  which  becomes  more  severely  felt  as  the  thickness  increases." 
In  proof  of  this  he  then  gives  the  elongation  of  holes  at  the  fracture,  in  a  series  of  tests  of  i",  i",  |",  \",  and 


APPENDIX. 


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4S»  •  APPENDIX. 

Table  IV. 

TESTS  OF  MEDIUM  STEEL  AND  HARD  STEEL. 


Test  cut  from. 

Specimen  Test. 

Punched  Piece. 

Reamed  Piece. 

E.  L. 

Ult. 

Elong. 

E.  L. 

P.  Ct. 

Ult. 

P.  Ct. 

E.  L. 

P.  Ct. 

Ult. 

P.  Ct 

Medium 
Steel, 
Riveted 
Holes 

8"  X  A"  P'ate 

T  1 "         A ' '  r\  1  'J  1  *a 

14    A  7  pidie 
loi"  X  14"  plate 
6"  X  6"  f"  angle 

39.460 
40,090 
38,400 

69,280 
67,600 

68,540 
66,340 
67,070 

20.25 
20  50 
29  38 
25-38 
23.00 

51,080 
45.550 
48,900 
45.970 
45,100 

109 
124 
115 
118 

70,460 
66, 1  go 
56,690 
23,300 
52,300 

102 

98 
83 

80 

78 

57,010 
53.460 
41,720 
46,990 
47.450 

128 
106 
118 
124 

72,700 
72,180 
71,630 
68.550 
66,840 

105 
107 
105 
103 
99 

Mean 

88.2 

Mean 

103.8 

Av  elong.  in 

8'  =4.70  p.  ct. 

Av.  elong.  in  8  '  =  8.75  p.  ct. 

Medium 
Steel. Open 
Holes 

7"  X  1"  plate 
6"  X  6"  X  i"  angle 

41,190 
33,010 

61,080 
66, 790 

19.50 

23-75 

40,610 
33,950 

99 
103 

54.260 
63.540 

89 
95 

45.420 
35.710 

no 
108 

58,980 
66,740 

97 
104 

Mean 

92 

Mean 

100.5 

Av.  elong.  in 

8"  =  8.5o  p.  ct. 

Av.  elong  in 

8"=  12 

D.  Ct. 

HardSteel, 
Open 
Holes 

15"  X  1"  plate 
*7"  X  piate 

48,380 
44,890 

8r,i20 
73.040 

22.25 
22.75 

55.430 
46,000 

115 
102 

57-420 
55,200 

71 

69 

67,500 
43,770 

139 

98 

71,210 

98 

Mean 

70 

This  specimen  had  a  flaw  in  it.    Form  of  specimens  same  as  in  Table  HL 


f"  plates,  as  follows:  With  drilled  holes  5.7,  14.1,  17.2,  18.7,  and  19.0  per  cent,  with  fractures  all  silky;  with 
punched  holes  3.2,  9.6,  13.5,  3.2,  and  1.9  per  cent,  with  the  last  two  fractures  showing  95  and  100  per  cent 
granular  respectively.  Then  he  anneals  the  specimens  in  a  similar  series,  to  remove  the  hardening  effect, 
and  gets  elongations  as  follows:  With  drilled  holes  "  extensions  of  11. 5,  16.3,  19.4,  21.0,  and  219;"  with 
punched  holes  "  10.3,  14.6,  18.  i,  19.3,  and  20.7  per  cent,  all  fractures  being  wholly  silky."  This  makes  a  very 
pretty  demonstration. 

Kirkaldy  compares  the  ultimate  strength  of  drilled  and  punched  specimens  in  these  same  tests,  as 
follows:  When  not  annealed,  the  strength  of  punched  pieces  as  compared  with  drilled  ones  showed  losses 
of  7.92,  8.10,  7.53,  23.45,  and  26.22  per  cent  respectively.  When  he  anneals  the  specimens,  however,  the 
punched  pieces  show  losses  of  but  5.85,  6.70,  6.74,  7.08,  and  7.18  per  cent  respectively.  "  This  manipulation 
or  treatment  rectifies  the  injuriousness  of  punching  to  a  great  extent,"  he  says.  This  is  argument  for  con- 
clusion No.  4,  because,  evidently,  so  far  as  the  rivets  anneal  and  soften  the  metal  they  benefit  it  and  lessen 
the  injury  in  punching;  and,  evidently  also,  this  advantage  accrues  more  to  thick  metal  than  to  thin 
because  the  original  injury  was  greater.  The  tables  in  this  paper  appear  to  offer  evidence  that  this  beneficial 
effect  of  riveting  is  an  actual  fact;  certainly,  the  riveted  pieces  show  higher  percentages  than  the  pieces 
with  open  holes.*  Regardless  of  any  such  effect,  however,  the  fact  that  punching  is  more  injurious  in  thick 
material  makes  it  desirable  to  use  moderate  sections  in  tension  members  and  in  floor-beams,  stringers,  and  girders 
subject  to  impact. 

In  concluding  the  discussion  of  the  tables,  attention  is  called  to  the  elongation  of  the  punched 
steel  pieces  in  a  length  of  8  inches  The  stretch  is,  of  course,  necessarily  confined  chiefly  to  the  three  holes  ; 
but  the  percentages  are  given  in  8-inch  length,  to  show  what  it  might  be  expected  to  average  in  any  long 
riveted  member. 

Having  thus  reasonably  demonstrated  more  than  an  equal  standing  for  our  punched  steel,  we  could 
logically  claim  for  it  unit  stresses  from  12  to  15  per  cent  higher  than  for  wrought-iron.  In  the  specifications, 
however,  it  was  thought  well  to  be  conservative  on  this  point,  especially  as  it  is  not  necessary  to  figure  soft 
steel  so  much  above  wrought-iron  in  order  to  make  the  cost  practically  the  same.  The  unit  stresses, 
therefore,  for  soft  steel  in  tension  are  placed  at  8  per  cent  higher  than  wrought-iron,  and  in  compression  10 
per  cent  higher.  These  figures  are  conservative,  and  the  remainder  is  so  tuuch  advantage  to  the  bridge 
structure. 


*  This  is  probably  due  more  to  the  frictional  resistance  to  movement  under  the  rivet  heads  than  to  the  annealing 
action  of  the  hot  nvet. — J.  B.  J. 


STRUCTURAL  STEEL  AND  GENERAL  SPECIFLCATLONS.  483 


It  will  be  observed  that  web-plates  are  required  to  be  of  steel  in  all  cases.  It  is,  of  course,  impracticable 
to  dimension  web-plates  for  unit  stresses.  It  would,  however,  be  absurd  to  use  steel  angles  in  connection 
with  an  iron  web,  since  the  equal  strength  of  steel  plates  in  all  directions  exactly  fills  the  condition  of  a  web- 
sheet,  and  makes  them  particularly  desirable  for  such  purposes.  There  is,  besides,  good  reason  for  throwing 
iron  web-plates  out  altogether,  because  of  poor  quality.  It  is  doubtless  no  longer  profitable  to  make  a  first- 
class  iron  web-plate;  hence  many  of  those  now  sold  are  of  very  doubtful  quality,  and  owe  whatever  virtues 
they  possess  to  the  percentage  of  steel  which  they  contain.  It  will  be  noted  that  both  iron  eye-bars  and  soft 
steel  eye-bars  are  ruled  out  of  the  specifications.  Excepting  with  the  most  conservative,  the  iron  bar  has 
already  been  practically  superseded,  because  steel  bars  are  now  cheaper  and  have  besides  abundantly 
demonstrated  their  superiority.  As  regards  soft  steel  bars,  there  is  no  reason,  on  the  face  of  it,  why  soft 
steel  should  not  make  a  good  eye-bar  •  it  doubtless  would  ;  but  medium  steel  has  been  found  10  answer  the 
requirements  with  entire  satisfaction,  and,  after  annealing,  is  considerably  softer  than  when  tested  in  its 
natural  state  in  small  specimens. 

There  is  a  practical  reason  also  for  not  using  soft  steel  in  the  fact  that  it  is  more  apt  to  have  piping  and 
blow-holes  in  it  than  medium  steel.  This  is  not  oljjectionable  in  angles  or  plates  where  these  defects  close 
up  and  disappear;  but  it  is  objectionable  in  large  masses,  like  pins  and  forged  bars.  It  was  not  thought 
worth  while,  therefore,  to  introduce  a  soft  steel  eye-bar  in  the  specifications.  The  result  is  that  the  soft 
steel  is  limited  to  compression  members,  to  girders,  and  to  riveted  tension  members,  and  the  idea  in  fixing 
the  unit  stress  has  been  to  place  it  for  these  purposes  on  a  par  with  other  material ;  possibly  to  give  it  a 
little  advantage  to  overcome  the  inertia  of  manufacturers  who  have  been  accustomed  to  using  wrought-iron 
only.  It  is  possible  that  it  is  not  necessary  to  make  as  much  allowance  as  has  been  made,  or  perhaps  it 
should  be  increased  to  suit  different  cases. 

Screw  Threads  on  Steel  Bars. 

There  is  one  point  more  that  may  require  special  mention — tiiat  is,  the  provision  of  the  specifications 
permitting  steel  bars  to  be  ust^d  with  upset  screws  on  their  ends.  This  is  probably  the  last  point  that 
engineers  are  ordinarily  willing  to  concede  in  regard  to  steel :  that  is,  the  possibility  of  putting  a  screw- 
thread  on  it  ;  nevertheless  this  may  be  done  with  entire  success,  and  in  evidence  thereof  the  following  tests 
are  submitted : 


Table  V. 

TESTS  AT  EDGE  MOOR  BRIDGE  WORKS  OF  STEEL  EYE-BARS  WITH  UPSET  SCREW  ENDS.* 


Eye  bars. 

Size. 

Length. 

Klastic  Limit. 

Ultimiite 
SirenEth 

Elongation, 
per  cent. 

Reduction,  per 
cent. 

Reinaiks. 

No.  1 
No.  2 
No.  3 

5"  X  V 
4"  X  I" 
5"  X  f 

37'  i" 
31'  6" 
37'  o|" 

39.370 
37.700 
38,170 

61,620 
59,  240 
62,470 

12.66  (33') 
15-63(27') 
11-71  (33') 

48.08 
51-42 
43-48 

Broke  in  long  end 
Broke  in  short  end 
Broke  in  long  end 

These  tests  are  certainly  crucial  tests,  as  it  would  be  impossible  to  imagine  a  more  diflicult  upset  than 
to  put  a  round  sctew  end  on  a  5  x  bar. 

Not  long  ago  one  of  the  leading  Southwestern  railroads  desired  to  renew  a  large  number  of  hangers, 
which  were  necessarily  litnited  in  size,  because  of  the  place  they  had  to  fit.  In  this  connection  Mr. 
Hetiry  G.  Morse,  and  President  of  the  Edge  Moor  Bridge  Works,  had  some  experimental  tests  of  steel 
hangers  made.    The  results  of  these  tests  are  given  in  Table  VI. 

The  material  that  was  put  in  the  hangers  was  taken  frotii  stock,  being  ordinary  steel  rounds  which 
had  been  ordered  for  cotter  pins.  Attention  is  called  to  the  fact  that  the  excess  in  areas  at  the  root  of  the 
thread  on  these  hangers  was  but  7  per  cent,  and  it  may  well  be  doubted  whether  iron  hangers  could  have 
stood  as  severe  a  test. 

The  tests  which  have  been  presented  show  that  the  steel  held  its  value,  after  punching,  better  than  iron, 
up  to  say  I"  thickness,  when  the  percentages  in  steel  and  in  iron  were  just  about  equal.  But  these  tests 
also  show  that  the  value  of  the  steel  decreased  quite  rapidly  as  the  thickness  increased  ;  the  percentages 
running  down  from  about  100  at  %"  thickness  to  75  at  f  thickness. 

In  iron  the  percentages  seemed  to  be  much  more  constant,  the  range  being  only  from  82  to  72  per  cent. 
(See  Tables  III  and  IV.)  In  order  to  make  this  fact  more  clearly  apparent,  the  results  are  plotted  as  a  diagram, 


This  table  is  of  little  value  without  some  knowledge  of  the  relative  net  areas  on  screw  and  bar  portions. 


484  APPENDIX. 

Table  VI. 

TESTS  AT  EDGE  MOOR  BRIDGE  WORKS  OF  STEEL  STIRRUP  HANGERS  AND  UPSET  RODS, 


Stirrup 

Diameter 

Diam.  at 
Root  of 
Thread. 

Area. 

Elastic 

Hangers. 

of  Bar. 

Orig. 

Fract. 

Limit. 

■  No.  I 

1-75 

1. 81 

2.41 

1.23 

38,540 

No.  2 
No.  3 
Upset  Rods 
No.  I 
No.  2 
No.  3 

1-75 
1-75 

1-75 
1-75 
1-75 

I. 81 
1. 81 

1. 81 
I. 81 
1. 81 

2.41 
2.41 

2.41 
2.41 
2.41 

1 .09 
•95 

1. 91 
2.06 
•99 

37,760 
37,760 

34.610 
40,900 
37,760 

Ultimate 
Strength. 

Elong.  in 
12",  per 
cent. 

59.780 

26.2 

58,990 

20-33 

58 

1517 

55 

16.67 

56 

II  .67 

5 

33^33 

Reduction 
of  Area, 
per  cent. 


48.96 

54-72 
60.58 


20.75 
14.52 
58.92 


Remarks. 


j  Broke  in  one  leg. 

I  i'  3^"  from  end. 
Broke  in  bend. 
Broke  in  bend. 

Broke  2'  gj^"  from  end. 
Broke  2'  si"  from  end. 
Broke. 


in  Fig.  451,  which  shows  the  percentage  values  of  the  punched  iron,  the  punched  steel  with  open  holes,  and 
the  punched  steel  with  riveted  holes  for  different  thicknesses  of  metal.    It  also  shows  a  mean  line  for  steel. 

In  addition  to  this  decline  in  percentage  value,  there  was  a  gradual  change  in  the  character  of  the 
fracture  from  fine  silky  in  f"  material  to  100  per  cent  granular  in  the  |"  material.  This  gradual  change  is 
shown  by  the  photograplis  in  Plate  XXXVIII,  which  indicate  how  the  granular  structure  first  appears  at  the 
edge  of  the  holes  and  radiates  from  them  in  larger  percentages  as  the  thickness  increases.  In  order  to  show 
the  latter  characteristics  more  fully,  Table  VII,  is  inserted,  which  gives  the  details  for  most  of  the  tests  of  soft 


Thickness  of  Metal 


Fig.  451. 


and  medium  steel  which  appear  in  Tables  III  and  IV.  The  tests  of  iron  did  not,  of  course,  show  granular 
fractures,  since  the  normal  structure  of  iron  is  not  granular,  as  that  of  steel  is.  But  it  was  apparently  very 
dead,  and  it  would  be  difficult  to  say  which  of  the  two  metals  was  more  reliable  at  \"  thickness.  The  writer 
is  disposed  to  think,  however,  that  tiiis  decrease  in  value  and  increase  in  granular  fracture  in  thick  steel 
should  be  considered,  regardless  of  a  satisfactory  comparison  in  ultimate  strength  with  wrought-iron  of  the 
same  gauges.    He  has  decided,  therefore,  to  limit  the  thickness  of  punched  material  in  the  specifications, 


PLATE  XXXVIII. 


Partly  Granular 


All  Granular. 


Norn. — See  also  cms  showing  effects  of  shearing  ani]  punching  solt-steel  plates,  Ktigxurfrin!^  .Wiw, 
May  2,  l8g5,  vol.  xxxiii.  p.  ago. 


Showing  the  Effects  of  Punching  Mild-Stccl  Plates  of  Diff  erent  Thicknesses. 


SCAI.K   1111,1,  SIZK. 


STRUCTURAL  STEEL  AND  GENERAL  SPECIFICATIONS.  485 


and  to  do  this  for  most  members  by  drawing  the  line  at  the  gauge  in  which  the  crystalline  fracture  becomes 
evident  and  the  stretching  qualities  decline.  From  Table  VII  it  is  apparent  that  all  the  tests  are  satisfactory 
until  we  pass  thickness.  Above  that  gauge  there  is  but  one  test  which  would  pass  muster  on  the  basis 
here  proposed.  A  third  condition  to  the  use  of  soft  steel  has  therefore  been  added,  limiting  to  \  inch  the 
thickness  of  material  subjected  to  punching,  excepting  that  in  girders  over  50  feet  long  it  may  be  in  top 
chords  and  end  posts  it  may  be  f",  and  in  shoes,  pedestals,  and  bed-plates  it  may  be  f". 

Table  VII. 


DETAILS  OF  PUNCHED  TESTS  GIVEN  IN  TABLES  III  AND  IV  FOR  SOFT  AND  MEDIUM  STEEL. 


Thick- 
ness. 

Specimen  cut  from 

Elastic 
Limit, 
lbs.  per 
sq.  in. 

Ultimate 
Strength, 
lbs.  per 
sq.  in. 

Stretch 
in  3  in., 
per  cent. 

Stretch 
of  Hole, 
per  cent. 

Character  of 
Fracture. 

Class  o{  Steel. 

i" 

7 

X  r_ 

Bess. 

58,840 

64,230 

8.50 

46 

9 

Silky 

Soft  steel 

Riveted  holes 

5  " 

8" 

X  i-^^^ 

(( 

50,780 

59,600 

8.00 

40 

8 

1' 

8" 

X  T  J 

51,080 

70,460 

5-75 

29 

6 

Medium  steel 

15" 

X  i" 

51,620 

61,220 

7.50 

36 

I 

Soft  steel 

1" 

4"  X  3" 

7" 

X  1" 

38,170 

50,330 

7-25 

Open 

r 

X  1" 

(( 

40,610 

54,260 

9- 50 

Medium  steel 

"  " 

i" 

12" 

X  4" 

47.450 

48,960 

4-75 

34 

0 

20  per  ct.  gran. 

Soft  sleel 

i" 

14" 

X  4" 

X  v;^ 

45.550 

66,igo 

7.00 

32 

6 

Silky 

Medium  sleel 

Riveted  " 

4" 

6"  X  6" 

0.  H. 

33.950 

63.540 

7  50 

Open  " 

9 " 

19V' 
12" 

X  Tff 

Hess. 

33,950 

52,400 

7- 50 

48 

9 

Silky 

Soft  sleel 

9  '' 

1  9" 
'TS 

X  TS, 

0.  H. 

35.310 

58,890 

8.75 

Granular 

18" 

X    3  ^ 

Bess. 

45,080 

.38 

3 

09 

1" 

I4i" 
loi" 

X  f" 

43,820 

.62 

4 

25 

2  1" 
¥5 

V   2  1' 

0.  H. 

48,900 

56,690 

1-75 

7 

22 

Medium  steel 

Riveted  " 

1 1 " 

Tff 

f  ' 

8" 

X  H' 

Bess. 

44,170 

45.500 

3.50 

23 

4 

70  per  ct.  gran. 

Soft  sleel 

Open  " 

18" 

X  f" 
X  f" 

40,070 

41,180 

•75 

6 

38 

Granular 

f" 
f" 

6"  X  6" 

0.  H. 

45.100 
45,970 

52,300 
53,300 

1.50 
1-35 

7 
5 

22 
15 

Medium  sleel 

Riveted  " 

SPECIFICATIONS  FOR  FIRST-CLASS  BRIDGE  SUPERSTRUCTURE.* 

General  Description. 

1.  All  parts  of  the  structure,  except  ties  and  guard-rails,  and  bed-plates  of  stringers,  shall 
be  of  wrought-iron  or  steel.  Stringer  beds  will  be  cast-iron.  Cast-iron  or  steel  for  other 
purposes  will  only  be  used  for  special  conditions  at  the  fiiscretion  of  the  Chief  Engineer. 

2.  Ties  and  guard-rails  shall  be  of  wood.  They  will  be  supplied  by  the  railway  company, 
but  must  be  put  in  place  by  the  contractor,  who  will  also  furnish  all  bolts,  nuts,  washers,  etc., 
(except  rail  spikes),  for  this  purpose. 

Ties  on  tangents  will  be  8  inches  by  10  inches,  laid  on  8-inch  face  and  spaced  14  inches 
between  centres;  guard-rails  will  be  7  inches  by  8  inches,  spaced  8  feet  between  centres. 

On  curves  the  outer  rail  will  be  elevated  \  inch  per  degree,  and  this  elevation  will  be 
framed  in  the  ties,  as  no  shims  will  be  allowed. 

Ties  will  be  notched  |  inch  over  stringers,  and  guard-rails  }  inch  over  ties. 

Guard-rails  will  be  bolted  to  each  end  of  every  other  tie;  and  ties  and  guard-rails  will  be 
secured  to  stringers  by  hook  bolts  at  each  end  of  every  fourth  tie. 

3.  For  spans  of  16  feet  or  less,  rolled  beams  will  be  used,  and  from  16  feet  to  100  feet, 
riveted  plate  girders.    All  spans  over  100  feet  will  be  pin-connected  trusses.t 

4.  Beams  or  deck  girders  on  masonry  will  be  spaced  7  feet  o  inches  centre  to  centre  (ties 
lo  feet  long).  Riveted  stringers  on  through  single-track  bridges  and  on  tre.stles  will  be 
spaced  9  feet  centre  to  centre  (ties  12  feet  long),  beam  stringers  8  feet  6  inches  on  double- 
track  through  bridges  7  feet  centre  to  centre,  symmetrically  disposed  under  the  rails. 

Tracks  will  be  13  feet  apart  centre  to  centre. 


Materials. 


Ties  and  guard- 
rails. 


superstructure. 


Stringer-spacing. 


*  Compiled  by  F.  H.  Lewis,  C.E. 


t  See  Chap.  XVI. 


486 


APPENDIX. 


Clearance  for 
through  spans. 


Trusses  on 
curves. 


Deck  trusses. 


Through  girders. 


Girders  on 
curves. 


Design. 


Collision  strut. 


5.  Through  bridges  on  tangents  shall  not  be  less  than  14  feet  in  width  in  the  clear  be- 
tween trusses  for  single  track,  and  27  feet  for  double  track,  nor  less  than  20  feet  in  height  in 
the  clear,  measuring  from  base  of  rail  to  the  lowest  point  of  portals. 

6.  On  curves  the  truss  on  the  convex  side  will  be  7  feet  from  the  centre  line  at  the  middle 
of  the  span  ;  at  the  ends  of  the  span  the  truss  on  the  inner  side  of  curve  will  be  spaced  7  feet 
+  o.2d  feet  from  the  centre  line,  d  being  the  degree  of  curve. 

The  case  of  a  curve  near  enough  to  a  bridge  to  require  elevation  of  the  rail  will  also  be 
considered  and  provided  for. 

7.  Deck  truss  bridges  on  tangents  will  be  spaced  at  least  10  feet  centre  to  centre  of 
trusses. 

8.  Through  plate  girders  will  be  spaced  at  least  13  feet  6  inches  centre  to  centre  on 
tangents. 

9.  Stringers,  deck  girders,  and  deck  truss  spans  on  curves  will  be  spaced  as  may  be  neces- 
sary to  suit  the  circumstances  and  to  satisfy  the  Engineer. 

10.  All  structures  will  be  simple  in  design,  and  admit  of  accurate  calculation  of  the 
stresses  in  each  member. 

11.  Pin  trusses  will  be  in  all  ordinary  cases  of  the  single-intersection  type  with  leaning 
end-posts. 

12.  All  end-posts  of  through-pin  spans  will  be  braced  by  collision  struts,  and  the  end- 
panels  of  the  bottom  chord  and  the  vertical  suspenders  will  be  stiff  members. 


Proposals. 


Plans,  etc. 


Stress  sheets. 


Grade  of  material 
used. 


Approval  of 
plans  and  details. 


13.  Proposals  for  bridge-work  will  be  submitted  on  invitation  of  the  Chief  Engineer,  and 
must  conform  with  these  specifications,  with  general  plans  or  descriptions  furnished  by  the 
Engineer,  and  with  other  conditions  provided  for  in  the  letter  of  invitation. 

14.  Complete  strain  sheets,  general  plans  of  structure,  and  detail  drawings,  shall  be  fur- 
nished to  the  Chief  Engineer  of  the  railway  company  without  charge. 

The  stress  sheets  must  show  for  each  member  the  maximum  stress  or  stresses  caused  by 
the  dead  load,  the  live  load,  and  the  wind  separately;  the  unit  stress  and  the  dimensions  and 
areas  of  cross-sections.  Also  the  dead  weight  assumed  in  the  calculation,  which  must  not  be 
less  than  the  actual  weight  of  the  structue  as  built. 

15.  It  will  be  noted  that  the  specifications  below  provide  as  follows ; 

(1)  That  all  eye-bars  and  pins  shall  be  of  medium  steel  (see  §§  133  and  146). 

(2)  That  all  web-plates  shall  be  of  steel  (see  §  76). 

(3)  That  loop  rods  and  all  other  devices  which  are  welded  shall  be  of  wrought-iron. 
These  requirements,  as  defined  below,  are  common  to  all  bridges,  whether  built  of 

wrought-iron,  soft  steel,  or  medium  steel.  The  other  parts  of  bridges,  however,  may  be  built 
of  such  grades  of  material  as  the  contractor  may  elect,  provided  only  that  each  member  and 
each  set  of  members  performing  similar  functions  must  be  of  the  same  grade  of  material 
throughout. 

16.  Complete  detail  drawings  must  be  submitted  for  approval  to  the  Chief  Engineer,  and 
work  will  not  be  commenced  until  the  stresses  and  details  have  been  approved.  The  Chief 
Engineer  or  such  assistants  as  he  may  appoint  shall  have  free  access  at  all  times  to  the  work- 
ing drawings  and  shops  of  the  contractor  for  the  purpose  of  examining  the  plans  and 
inspecting  the  material  used  and  the  mode  of  construction. 

17.  The  contractor  shall  furnish  to  the  Chief  Engineer  of  the  railway  company,  free  of 
cost,  such  detail  drawings  of  each  structure  as  he  may  require. 


Loading. 
Live  Loads. 

18.  Live  loads  will  be  as  per  diagram  furnished  by  the  Chief  Engineer. 

19.  The  structure  will  be  proportioned  to  carry  the  live  loads  as  per  diagram,  and  the 
live  load  stresses  will  be  the  maximum  stresses  produced  by  the  rolling  load  considered  as 
stationary  or  as  moving  in  either  direction.  In  double-track  structures,  one  track  or  both 
will  be  considered  loaded,  whichever  may  produce  the  greater  stresses,  and  the  trains  will  be 
supposed  to  move  either  in  the  same  or  in  opposite  directions. 


STRUCTURAL  STEEL  AND  GENERAL  SPECIFICATIONS.  487 


Dead  Load. 

20.  The  dead  load  shall  consist  of  the  entire  structure,  including  the  floor  system  and 
rails  and  fastenings.  The  weight  of  the  ties,  guard  timbers,  rails,  spikes,  etc.;  shall  be  taken 
at  400  lbs.  per  lineal  foot  for  each  track. 

The  load  of  the  structure  when  complete  shall  not  exceed  the  dead  load  used  in  calculat- 
ing the  stresses. 

21.  In  through  bridges,  two  thirds  (f)  of  the  dead  load  shall  be  assumed  as  concentrated 
at  the  joints  of  the  bottom  chord,  and  one  third  (^)  at  the  joints  of  the  upper  chord. 

In  deck  bridges,  two  thirds  (f)  of  the  dead  load  shall  be  assumed  as  concentrated  at  the 
joints  of  the  upper  chord,  and  one  third  {\)  at  the  joints  of  the  bottom  chord. 

Wind  in  Trusses. 

22.  The  bottom  lateral  bracing  in  deck  bridges  and  the  top  lateral  bracing  in  through 
bridges  must  be  proportioned  to  resist  a  uniformly  distributed  lateral  force  of  150  lbs.  per 
lineal  foot  of  bridge  for  all  spans  of  200  feet  and  under,  and  an  additional  force  of  10  lbs.  per 
lineal  foot  for  every  25  feet  increase  in  length  of  span  ove  200  feet. 

23.  The  bottom  lateral  bracing  in  through  bridges  and  the  top  lateral  bracing  in  deck 
bridges  must  be  proportioned  to  resist  a  uniformly  distributed  force  the  same  as  above,  and 
an  additional  force  of  300  lbs.  per  lineal  foot  of  bridge,  which  will  be  treated  as  a  moving 
load. 

Wind  in  Trestles. 

24.  Trestles  shall  be  so  proportioned,  and  the  trestle  bents  shall  have  such  a  spread  of 
base,  that  no  tension*  may  occur  in  the  windward  trestle-leg  when  the  structure  is  loaded 
with  a  light  train  weighing  600  lbs.  per  lineal  foot  of  track,  and  when  the  wind  pressure  on 
this  train  is  300  lbs.  per  lineal  foot,  acting  9  feet  above  the  rail.  In  addition,  the  wind 
pressure  on  the  structure  itself  shall  be  assumed  at  not  less  than  150  pounds  for  each  longi- 
tudinal foot  of  structure,  and  not  less  than  100  pounds  for  each  vertical  foot  of  height  of  each 
trestle  bent,  and  more  if  the  exposed  wind  surface  of  track  and  structure  exceeds  5  square  feet 
for  each  foot  in  length,  and  the  exposed  wind  surface  of  each  trestle  bent  exceeds  3^  square 
feet  for  each  vertical  foot  of  height. 

In  all  cases  the  projected  surface  of  both  sides  of  the  towers  and  of  one  train  is  to  be 
taken  as  the  surface  acted  upon  by  the  wind. 

Centrifugal  Force. 

25.  When  the  bridge  is  on  a  curve,  add  to  the  maximum  wind  stresses  a  moving  lateral 
stress  equal  to  3  per  cent  of  the  live  load  on  all  tracks  (acting  in  the  direction  of  centrifugal 
fofce)  for  each  degree  of  curvature. 

26.  The  effects  of  wind  and  centrifugal  force  in  the  lateral  system  of  structures  must  be 
fully  provided  for  at  unit  stresses  given  below. 

Longitudinal  Bracing  and  Anchorage. 

27.  Longitudinally  the  bracing  of  trestle  towers  and  the  attachments  of  the  fixed  ends  of 
all  trusses  shall  be  capable  of  resisting  the  greatest  tractive  force  of  the  engines  or  any  force 
induced  by  suddenly  stopping  the  assumed  maximum  trains,  the  coefficient  of  friction  of  the 
wheels  upon  the  rails  being  assumed  to  be  0.20. 

Double-track  structures  will  be  braced  to  provide  for  trains  moving  either  in  the  same  or 
in  opposite  directions. 

Temperature. 

28.  Variations  in  length  from  change  of  temperature  to  the  amount  of  i  inch  in  100  feet 
shall  be  provided  for. 

Calculations  and  Unit  Stresses. 

29.  All  parts  of  the  structure  will  be  proportioned  to  sustain  the  maximum  stresses  pro- 
duced by  the  live  and  dead  loads  specified  above,  and  by  the  wind  and  centrifugal  forces 
under  special  conditions  provided  in  paragraphs  26,  32,  and  39. 

30.  In  calculating  stresses,  conventional  assumptions  will  be  used  throughout.  The 
Assumed  spans,   lengths  of  spans  will  be  the  distance  between  centres  of  end-pins  of  trusses,  and  between 

centres  of  bearing  plates  of  beams  and  girders. 


[*  An  unnecessary  limitation  in  high  trestles. — J.  B.  J.] 


488 


APPENDIX. 


Assumed  lengths.        The  length  of  Stringers  will  be  the  distance  between  centres  of  floor-beams,  and  the 
length  of  floor-beams  the  distance  between  centres  of  trusses. 

The  depth  for  calculation  of  girders  will  be  the  distance  between  centres  of  gravity  of 
flange  sections,  provided  it  does  not  exceed  the  distance  out  to  out  of  angles,  in  which  case 
the  latter  amount  shall  be  considered  the  depth. 

The  length  of  posts  will  be  the  centre-to-centre  length  between  pins,  except  in  trestle- 
posts,  where  the  length  will  be  from  cap  or  base-plate  to  the  centres  of  intermediate  struts. 

In  estimating  the  section  of  the  end-posts,  the  collision-strut  connection  will  not  be  con 
sidered. 

Any  modification  of  this  will  be  at  the  discretion  of  the  Engineer. 

Bending  in  pins  and  rivets  will  be  estimated  between  centres  of  bearings. 

Formulae  for  Unit  Stresses. 

31.  The  following  formulae  for  unit  stresses  in  pounds  per  square  inch  of  net  sectional 
area  shall  be  used  in  determining  the  allowable  working  stress  in  each  member  of  the 
structure  : 

TENSION  MEMBERS. 


Wrought-iron. 

Soft  Steel. 

Medium  Steel. 

(a)  Floor-beam  hangers  or  suspenders. 

Suspenders,  hangers  and  counters, 
riveted  members.net  seciion(see 

§  140)  

(V)  Solid  rolled  beams  (by  moments  of 

(c)  Riveted  truss  members  and  ten- 
sion* flanges  of  girders,  net  sec- 

Will  not  be  used 
6,000 

5,000 
8,000 

/    .  min.  \ 

7,000  li  -\  1 

V  max./ 

Will  not  be  used 
15,000 

Will  not  be  used 
<(     <i    tt  << 

5,500 
8,000 

8  %  greater  than  iron 
Will  not  be  used 
16,000 

7,000 
7,000 

7,000 

Will  not  be  used 

/  min.V 
9,000  li-^  1 

/    ,  min.V 

9,000  liH  1 

\  max./ 

/For  eye-bars  onlyV 
\         17,000  / 

COMPRESSION  MEMBERS. 

Wrought-iron. 

Soft  Steel. 

Medium  Steel. 

(e)  Chord  sections: 

Chords  with  pin-ends  and  all  end- 

(A)  Lateral   struts,  and  compression 
in  collision  struts,  stiff  suspen- 

/    ,  min.V  / 

7,000  ll  A  1  —  30- 

\  ^  max./      ^  r 

I    .  min.V  / 

7,000  li-^  1—  35  - 

\       max./  r 

I    ,  min.V  / 

7,000  li  -i  1-  40  - 

\       max./  r 

/    ,  min.V  / 

7,000   1-  35  - 

V       max.'  r 

I 

7,500  —  40  - 
/ 

10,500  —  50  - 
r 

t 

10  %  greater  than  iron 
<<       (<         tt  tt 

tt       tt         tt  tt 

tt       tt         tt  tt 

tt       tt         tt  tt 

(t       tt         tt  tt 

20  %  greater  than  iron 

<«              4*                   €4  it 

t*            *t                 tt  tc 
it             *t                   tt  tt 
tt             tt                 ti  tt 
tt            tt                 tt  tt 

In  these  formulae,  /  =  length  of  compression  member  in  inches,  and  r  =  least  radius  of  gyration  of  member  in 
inches.  If  the  allowable  stresses  given  by  formula  (c)  are  less  than  those  given  by  {e),  (/),  and  (g),  the  former  will  be  used. 


*  The  compression  flanges  of  beams  and  plate  girders  will  have  the  same  cross  section  as  the  tension  flange, 
f  Prof.  Burr  recommends  15  %  and  22  %  here  in  place  of  10  %  and  20  — J.  B.  J, 


STRUCTURAL  STEEL  AND  GENERAL  SPECIFICATIONS.  489 


MEMBERS  SUBJECT  TO  ALTERNATE  TENSION  AND  COMPRESSION. 


Wrought-iron. 

Soft  Steel. 

Medium  Steel. 

Use  the  formulae  above 

/        max.  lesser.  \ 

8  %  greater  than  iron 

20  %  greater  than  iron 

7,000  1 1  --I 

\       2  max.  greater/ 

Use  the  one  giving  the  greatest  area 

of  section 

COMBINED  STRESSES. 

(Jt)  When  the  cross-ties  rest  directly  on  the  top  chords,  the  latter  will  be  considered  as  beams  of  one  panel  length, 
subject  to  the  maximum  bending  that  will  result  from  the  wheel  loads  and  floor  system;  the  beam  to  be  considered  as 
supported  at  the  ends  for  section  in  centre  of  panel,  and  fixed  at  the  ends  for  section  at  the  ends  of  panel.  The  chords 
will  be  proportioned  to  sustain  the  algebraic  sum  of  the  stresses  resulting  from  the  direct  compression  or  tension  and 
the  transverse  loading  given  above,  in  which  the  allowed  stress  per  square  inch  shall  not  exceed: 


Wrought-iron. 

Soft  Steel. 

Medium  Steel. 

8,000 
10,000 

8,800 
11,000 

g,6oo 
12,000 

SHEARING. 

Wrought-iron. 

Soft  Steel. 

Medium  Steel. 

6,000 
4,800 
Will  not  be  used 

6,600 
5,200 
5,000 

7,200 
Will  not  be  used 
6,000 

BEARING. 

Wrought-iron. 

Soft  Steel. 

Medium  Steel. 

{nt)  On    projected    semi-intrados  of 

On  projected   semi-intrados  of 

Excepting  that  in  pin-connectea 
members  taking  alternate  stresses, 
the  bearing  stress  must  not  exceed 
9,000  pounds  for  iron  or  steel 

12,000 
12,000 

15,000 
250  lbs.  per  square  inch 

13,200 
<< 

16,500 

14,500 
<• 

18,000 

BENDING. 

Wrought-iron. 

Soft  Steel. 

Medium  Steel. 

(«)  On  extreme  fibres  of  pins  when 
centres  of  bearings  are  considered 
as  points  of  application  of  strains 

15,000* 

16,000* 

17,000* 

*  Prof.  Burr  would  use  the  Launhardt  Formula  here,  the  same  as  for  the  tension  and  compression  members. — J.  B.  J. 


490 


APPENDIX. 


{o)  Coefficients  ok  Friction  will  be  used  as  follows  : 

Wrought-iron  or  steel  on  itself  15 

"  "         cast-iron  20 

"  "         masonry  25 

Masonry  on  itself  50 


EffeCiS  of  wind  in 
truss  members. 


Collision  struts. 
Stiff  suspenders. 

Stiflf  chords. 


Bending  due  to 
weight  of  member. 


Flange  areas  of 
girders. 


Value  of  rivets. 


Deducting  rivet- 
holes. 


Minimum  com- 
pression members. 


Camber. 


Initial  stress. 


32.  In  case  the  maximum  stresses  in  chords,  girder  flanges,  trestle-posts,  or  the  bending 
effects  on  posts  due  to  wind  or  centrifugal  force  shall  exceed  25  per  cent  of  stresses  due  i< 
dead  and  live  load,  the  section  will  be  increased  until  the  total  stress  per  square  inch  will  not 
exceed  by  more  than  25  per  cent  the  maximum  fixed  for  live  and  dead  load  only. 

33.  Collision  struts  will  be  proportioned  to  carry  a  thrust  of  50.000  pounds  acting  in  a 
direction  at  right  angles  to  the  end-posts. 

Stifif  suspenders  must  be  able  to  carry  a  compressive  stress  equal  to  six  tenths  {^^)  the 
maximum  tensile  stress. 

T 

Stiffened  chords  will  be  proportioned  to  take  compression  equal  to  60—,  in  which  T'is 

the  maximum  tension  in  pounds  in  the  chord,  and  L  is  the  span  in  feet.  Use  formula  {h)  for 
these  members. 

34.  The  effects  of  the  weights  of  horizontal  or  inclined  members  in  reducing  their 
strength  as  columns  must  be  provided  for.  It  will  also  be  considered  in  fixing  the  position  of 
pin  centres. 

35.  Plate  girders  shall  be  proportioned  upon  the  supposition  that  the  bending  or  chord 
strains  are  resisted  entirely  by  the  upper  and  lower  flanges,  and  that  the  shearing  or  web 
strains  are  resisted  entirely  by  the  web  plate. 

36.  The  effective  diameter  of  the  driven  rivet  shall  be  considered  the  same  as  the 
diameter  before  driving. 

In  deducting  for  rivet-holes,  the  diameter  of  the  hole  will  be  considered  \  inch  greater 
than  the  rivet  for  full-headed  rivets,  and  \  inch  larger  for  countersunk  rivets. 

37.  No  compression  member  shall  have  a  length  exceeding  45  times  its  least  width,  and 

/ 

no  post  will  be  used  in  which  —  exceeds  125. 

38.  All  bridges  with  parallel  chords  shall  be  given  a  camber  by  making  the  panel  lengths 
of  the  toj)  chord  longer  than  those  of  the  bottom  chord  in  the  proportion  of  \  inch  to  every 
10  feet. 

39.  An  addition  of  10,000  pounds*  must  be  made  to  the  stress  obtained  in  all  lateral  rods 
to  provide  for  initial  tension,  and  the  proper  component  of  this  total  stress  is  to  be  used  in 
the  calculation  of  all  lateral  struts. 


Minimum  sections, 


Tension  bars. 


Details  of  Construction  and  Workmanship. 

i 

General. 

40.  All  details  must  be  of  approved  forms  and  satisfactory  to  the  Chief  Engineer. 

41.  Preference  will  be  had  for  such  details  as  will  be  most  accessible  for  inspection, 
cleaning,  and  painting. 

42.  No  shape  iron  weighing  less  than  six  pounds  per  lineal  foot  will  be  used,  nor  any 
iron  less  than  f  inch  thick,  nor  any  bar  of  less  than  one  square  inch  section. 

No  angle  smaller  than  3  inches  by  3  inches  will  be  used  in  girders  or  truss  members,  or 
in  any  member  having  |  inch  rivets. 

No  angle  smaller  than  2\  inches  by  2|  inches  will  be  used  in  any  part  of  bridge  struc- 
tures. 

End  angles  carrying  stringers  and  floor-beams  will  be  at  least  \  inch  thick  (see  §§  60  an. 

loi). 

43.  All  bed-plates  will  be  at  least  f  inch  thick. 

44.  No  tension  bar  will  be  less  than  i  inch  thick,  and  all  tension  bars  will  be  of  rectangu- 
lar section. 


[*  Whenever  the  working  stress  in  these  members  exceeds  twice  the  initial  stress,  the  initial  stress  in  the  counter- 
(rod  has  disappeared,  and  hence  does  not  need  to  b°  added.    See  footnote,  p.  228. — J.  B.  J.] 


STRUCTURAL  STEEL  AND  GENERAL  SPECIFICATIONS. 


491 


Pins. 


Universal  plates. 


Pitch  of  rivets. 


No  eye-bars  over  2  inches  thick  will  be  used ;  the  maximum  size  of  eye-bars  will  be  8 
inches  by  2  inches. 

45.  No  main  pins  will  be  less  than  3^  inches  diameter,  nor  less  in  diameter  than  three 
quarters  of  the  width  of  the  widest  bar  attaching  to  them. 

46.  Angles,  cover-plates  and  web  siieets  shall  be  as  long  as  practicable  to  avoid  splicing. 
Plates  for  all  purposes  shall  be  universal-rolled  when  the  width  does  not  exceed  30  inches 

(see  §§  76,  78.  124). 

47.  The  pitch  of  rivets  will  not  be  less  than  3  diameters,  nor  more  than  6  inches  (see  § 
92),  nor  more  than  16  times  the  thickness  of  the  thinnest  outside  plate.  No  rivet  will  have 
a  longer  grip  than  5  times  its  diameter,  nor  be  nearer  the  edge  of  the  metal  through  which 
it  passes  than  \\  inches  when  the  edges  are  machine-  or  roll-finished,  and  will  7tot  be  nearer 
than  if  inches  to  a  sheared  edge. 

48.  The  unsupported  width  of  any  plate  subjected  to  compression  shall  never  exceed  30 
times  its  thickness,  excepting  cover-plates  of  top  chords  and  end-posts,  where  it  may  be  40 
times  its  thickness. 

49.  Lattice-bars  will  be  as  described  in  §  122. 

50.  All  workmanship  must  be  first  class  ;  the  several  pieces  forming  a  built-up  member 
must  fit  snugly  together  without  open  joints.  This  will  be  specially  insisted  upon  with  refer- 
ence to  pill-bearing  chords  and  end-posts. 

All  members  must  be  straight  and  out  of  wind  ;  holes  accurately  bored  both  in  position 
and  direction  ;  lengths  correct  within  j'y  inch  in  all  cases  and  within  -^^  inch  for  all  pieces  of 
the  same  set. 

51.  Work  will  be  designed  in  clean,  handsome  lines.  In  addition  to  the  value  and  use- 
fulness of  members  in  the  structure,  they  will  be  neatly  finished. 

52.  All  material  must  be  straightened  before  being  laid  off,  and  also  after  punching,  if 
necessary. 


I-Beam  Spans. 


I-beam  stringers. 


Hitch  angles. 


Milled  ends. 


53.  I-beam  stringers  on  masonry  will  preferably  be  double  under  each  rail,  spaced  7  feet 
O  inches  between  centres  of  bed-plates. 

54.  They  will  have  planed  sole  plates  riveted  to  the  flanges,  and  bolted  through  bed- 
plates to  the  masonry  at  one  end,  and  free  to  slide  longitudinally  at  the  other.  They  will 
have  rigid  cross-struts  and  transverse  bracing  riveted  to  the  webs. 

55.  When  in  pairs,  they  will  have  wrought-metal  separators. 

56.  Bed-plates  will  be  cast-iron,  3"  high. 

57.  All  holes  in  flanges  will  be  drilled.    Holes  in  webs  may  be  punched. 

58.  I-beam  stringers  between  floor-beams  will  preferably  be  in  single  lines  under  each 
rail,  and  will  have  lateral  bracing  between  webs  for  all  spans  over  12  feet. 

59.  They  will  be  spaced  8  feet  6  inches  between  centres,  and  be  riveted  to  webs  of  floor- 
beams. 

60.  The  angles  carrying  them  at  the  ends  will  be  either  4  inches  by  4  inches  by  i  inch,  or 
6  inches  by  6  inches  by  \  inch. 

61.  If  resting  on  top  of  floor-beams,  they  will  be  riveted  to  the  floor  beams,  and  knee- 
braces  will  support  both  ends  of  each  line  of  beams. 

62.  On  the  abutments,  they  will  have  cross-struts,  and  be  stayed  'o  the  main  shoes  of 
girders  or  trusses.    Sole-plates  and  bed-plates  (see  §§  54  and  56). 

63.  Beams  fitting  between  floor-beams  must  be  milled  to  length  and  have  the  hitch  angles 
accurately  set,  square  to  the  stringer  and  flush  with  its  end. 

64.  Beams  which  are  not  required  to  be  of  exact  length  may  vary  \  incl»  from  dimensions 
called  for,  and  may  have  the  ends  cut  by  cold  saw. 

65.  Heating  beams  to  cut  them  will  not  be  allowed,  and  no  patched  beHi»is  wMi*  be  re- 
ceived. 


Rivets. 


66.  Rivets  will  be  'i  inch  or  \  inch  diameter  in  the  rod  before  upsetting. 

67.  For  main  members  |-inch  rivets  will  preferably  be  used,  with  f  inch  livets  for  lateral 
members. 


492 


APPENDIX. 


Steel  webs. 


Sti£Eeners. 


Fillers. 


Fit  of  stiffeners, 
ets.. 


Rivet  spacing  for 
local  shear. 


68.  Rivets  will  be  power-driven  wherever  practicable,  and  must  have  full  round  heads 
concentric  over  the  shank  of  the  rivet. 

69.  Whenever  the  grip  length  of  rivet  exceeds  24  inches,  power-driven  rivets  will  be  in- 
sisted upon. 

70.  All  rivet-holes  will  be  accurately  laid  ofi  and  punched. 

71.  The  dies  will  not  exceed  the  diameter  of  rivet  by  more  than  inch. 

72.  When  the  several  pieces  forming  a  member  are  bolted  up,  the  holes  must  match 
accurately  throughout,  or  the  material  may  then  be  condemned. 

73.  No  drifting  will  be  allowed  ;  holes  requiring  it  must  be  reamed;  hot  rivets  must  enter 
the  holes  without  the  use  of  a  hammer. 

74.  Countersinking  will  be  neatly  done;  all  holes  of  the  same  size,  and  countersunk  rivets 
must  completely  fill  the  holes. 

75.  Rivets  must  completely  fill  the  holes,  and  no  loose  or  badly  formed  rivets  will  be 
allowed,  nor  any  calking.    Field  riveting  must  be  reduced  to  a  minimum. 

Plate  Girders. 

76.  The  webs  of  all  plate  girders  will  be  of  steel  (see  Quality  of  Material,  §§  171-184). 
Universal  plates  will  be  used  for  widths  up  to  30  inches,  and  sheared  plates  for  greater 

widths. 

77.  No  rivet-holes  will  be  pitched  nearer  than  if  inches  to  a  sheared  edge,  or  nearer  than 
\\  to  a  roll-finished  or  machined  edge  of  the  webs. 

78.  Web  splices  will  be  made  by  two  universal  steel  plates  of  the  same  thickness  as  the 
web  plate. 

Stiffeners  will  be  used  at  all  web  splices. 

The  width  of  the  splice-plates  shall  be  sufficient  to  admit  the  requisite  number  of  rivets 
and  to  receive  the  stiffeners. 

79.  Wherever  the  unsupported  distance  between  tlie  flange  angles  exceeds  50  times  the 
thickness  of  the  web  sheet,  vertical  stifTeners  of  angle  iron  shall  be  placed  on  each  side  of  the 
girder. 

Stiffeners  will  be  symmetrically  spaced  from  the  centre  of  the  girder,  and  the  distance 
between  them,  centre  to  centre  of  rivets,  will  not  be  greater  than  the  distance  between  centres 
of  flange  angles.  If  unequally  spaced,  the  distance  between  them  will  decrease  toward  the 
ends. 

80.  There  will  be  a  pair  of  stiffeners  at  each  end  of  all  bed-plates. 

81.  All  stiffeners  will  have  fillers  under  them  of  the  same  thickness  as  flange  angles  and 
as  wide  as  stiffener  angles. 

82.  The  net  section  of  tension  flanges  will  be  reckoned  as  the  minimum  section  square 
across  the  flange,  and  the  net  section  on  any  diagonal  or  broken  line  through  two  or  more 
rivet-holes  must  have  25  per  cent  excess. 

83.  In  calculating  shearing  and  bearing  stresses  on  web  rivets  of  plate  girders,  the 

maximum  shear  acting  on  the  outer  side  MM  of  any   m  o 

panel  will  be  considered  to  be  transferred  to  the  flange 
angles  in  a  distance  MO  (equals  MM),  and  the  num- 
ber of  rivets  in  the  stiffener  MM  will  follow  the  same 
rule.* 

84  All  stiffeners,  fillers,  and  splice  plates  on  the 
webs  of  girders  must  fit  at  their  ends  to  the  flange 
angles  sufficiently  close  to  be  sealed,  when  painted, 
against  admission  of  water,  but  need  not  be  tool- 
finished. 

85.  Web  plates  of  all  girders  must  be  arranged  so 
as  not  to  project  beyond  the  faces  of  the  flange  angles, 
nor  on  the  top  be  more  than  jV  '"^h  below  the  face 
of  these  angles  at  any  point. 

86.  To  provide  for  local  shear  of  heavy  wheel  loads,  the  rivet  spacing  in  top  flanges  of 
deck-plate  girders  and  stringers  will  not  exceed  3  inches  pitch  when  there  are  no  cover  plates, 
or  4  inches  with  cover  plates. 


poo 


o  o 


000000000000^ 


oooooooo 


o  o  o\ 


Fig.  452. 


[*See  Art.  279  for  the  true  theory  and  practice. — J.  B.  J.] 


STRUCTURAL  STEEL  AND  GENERAL  SPECIFICATIONS. 


493 


Vateral  bracing. 


One  top  flange 
plate,  etc. 


Flange  splices. 


Cross-frames. 


Finish  of  girders. 


87.  The  compression  flanges  of  girders  will  be  stayed  at  intervals  not  exceeding  15  times 
their  width. 

88.  In  through  spans,  stiffened  gussets  will  run  from  top  flange  to  each  floor-beam. 

89.  In  deck  spans  the  lateral  bracing  will  extend  from  end  to  end,  and  no  brace  will 
make  an  angle  less  than  40  degrees  with  the  girders,  excepting  end  braces  of  skew  spans. 

90.  All  girders  having  flange  plates  will  have  one  plate  in  each  flange  extending  from 
end  to  end ;  and,  with  the  exception  of  floor-beams,  girders  will  preferably  have  at  least  one 
flange  plate. 

91.  When  two  or  more  plates  are  used  on  the  flanges  they  shall  either  be  of  equal  thick- 
ness or  shall  decrease  in  thickness  outward  from  the  angles,  and  shall  be  of  such  lengths  as 
to  allow  of  at  least  two  rows  of  rivets  of  the  regular  pitch  being  placed  at  each  end  of  the 
plate  beyond  the  theoretical  point  required. 

92.  When  two  or  more  cover-plates  over  12  inches  wide  are  used  in  the  flanges  of  plate 
girders,  an  extra  line  of  rivets  shall  be  driven  along  each  edge  to  draw  the  plates  together  and 
to  prevent  the  entrance  of  water. 

Plates  over  17  inches  wide  will  have  three  rows  of  rivets  with  9  inches  pitch  for  the  outer 

row. 

93.  All  joints  in  flanges,  whether  in  tension  or  compression  members,  must  be  fully 
spliced,  as  no  reliance  will  be  placed  upon  abutting  joints.  The  ends,  however,  must  be 
dressed  straight  and  true,  so  that  there  shall  be  no  open  joints. 

94.  Flange  angles  must  be  spliced  with  angle-covers  whenever  cut  within  the  length  of 
the  girder. 

95.  Splices  must  break  joints  with  each  other,  one  piece  only  being  spliced  at  any  point. 

96.  Cross-frames  will  be  used  at  the  masonry  ends  of  all  girders  and  at  intermediate 
points  when  wind  or  centrifugal  force  makes  it  desirable. 

97.  All  cross-frames  will  be  made  of  angles  and  plates,  and  will  be  stiff  rectangles  of  four 
members,  viz.,  top,  bottom,  and  two  diagonals. 

98.  Girders  will  be  neatly  finished  at  the  ends.  They  will  have  a  plate  corresponding  in 
width  with  the  cover-plates  riveted  to  end  stiffeners,  and  a  corner  cover  at  the  top  riveted  to 
both  top  and  end  plates. 


Stringers  and  Floor-beams. 


Steel  webs.  99-  Webs  of  Stringers  and  floor-beams  will  be  steel  (see  §  76). 

100.  Stringers  and  floor-beams  will  preferably  be  carried  by  rivets  at  their  ends,  and  all 
such  ends  will  be  milled  to  exact  length. 

101.  The  end  angles  carrying  floor-beams  and  stringers  will  be  either  4X4xi,  or6x4 
X  I,  or  6  X  6  X  ^  inch  angles,  as  one  or  two  rows  of  rivets  are  required,  and  they  will  have 
fillers  under  them  7  inches  and  9  inches  wide,  respectively,  of  same  thickness  as  flange  angles, 
and  having  a  row  of  rivets  outside  the  hitch  angles. 

102.  The  hitch  angles  of  stringers  will  have  a  full  complement  of  rivets,  and,  in  addition 
stringers  may  have  a  bracket  and  stiffeners  under  them  unless  they  practically  cover  the  web 
of  floor-beam  between  flanges. 

103.  All  holes  for  field  rivets  in  stringer  and  floor-beam  connections  will  be  punched  f  inch 
diameter,  and  afterward  reamed  out  to  \\  inch,  using  cast-iron  templates.  The  wire  edge  on 
reamed  holes  must  be  removed. 

104.  Floor-beams  having  stringers  resting  on  their  top  flanges  will  have  stiffeners  under 
the  points  of  support. 

105.  Stringers  will  preferably  have  one  cover-plate  from  end  to  end  (see  f  90). 

The  rivet  pitch  through  the  web  in  top  flanges  of  stringers  will  not  exceed  4  inches 
throughout  (see  §  86). 

Hangers.  106.  Hangers  will  be  either  solid  forged  eye-bars  riveted  to  floor-beams,  or  riveted  plate 

hangers.    Stirrup  hangers  or  bent  hangers  will  not  be  used. 

107.  Eye-bar  hangers  will  have  the  usual  excess  of  material  across  the  eye,  and  all  rivet- 
holes  in  them  will  be  bored,  and  the  edges  of  holes  will  be  chamfered  to  make  a  fillet  undei 
rivet-head. 

108.  All  riveted  hangers  will  have  excess  sections  at  the  nin-holes,  as  described  in  §  139 


Hitch  angles. 


Reamed  holes  for 
field  rivets. 


494 


APPENDIX. 


Long  Plate  Girders. 

109.  Plate  girders  of  lengths  trom  75  tt.  to  100  ft.  may  be  built  of  medium  steel,  under 
special  conditions,  as  follows  : 

no.  All  rivet-holes  in  the  angles  and  plates  in  both  flanges  will  be  drilled  in  the  solid. 
Occasional  small  holes  may  be  punched  for  purposes  of  bolting  up  and  reamed  afterward. 

111.  All  other  rivet-holes  in  web  plates,  stiflfeners,  lateral  braces,  and  fillers  may  be 
punched  of  full-size  for  riveting,  provided  the  metal  does  not  exceed      inch  thickness. 

112.  The  spliced  ends  of  tension  flange  plates  or  angles  will  be  milled  off  i"  to  remove 
sheared  edges  ;  and  the  ends  of  all  splicing  pieces  in  tension  flange  will  be  similarly  milled. 
No  other  milling  of  sheared  edges  will  be  required,  excepting  such  as  is  specified  for  iron 
girders. 

113.  All  web  splices  will  have  four  rows  of  rivets,  the  middle  rows  being  pitched  five 
inches  between  centres. 

114.  Rivets  will  be  of  soft  steel  and  power-driven  whenever  practicable. 

115.  The  general  requirements  given  under  Plate  Girders  will  apply  to  these  also,  except 
as  modified  above.    (See  §§  77  to  98  inclusive.) 


Pin-connected  Spans. 


Lattice. 


116.  See  §§  10,  II,  12,  31,  33,  and  47. 

117.  No  web  member,  except  collision  struts,  shall  make  a  less  angle  with  the  horizontal 
than  45  degrees. 

Compression  Members. 

118.  All  parts  working  together  as  one  member  shall  be  uniformly  stressed. 

119.  All  eccentricity  of  stress  shall  be  avoided.    Pin-centres  will  be  in  the  centre  of  grav- 
Centre  of  gravity,  ity  of  the  members,  less  the  eccentricity  required  to  provide  for  their  own  weight;  and  in 

continuous  chords,  pin-centres  must  be  in  the  same  plane. 

120.  See  §§  46  and  69. 

121.  The  open  sides  of  all  compression  members  shall  be  stayed  by  batten  plates  at  the 
Batten  plates,    ends,  and  diagonal  lattice-work  at  intermediate  points.    The  batten  plates  must  be  placed  as 

near  the  ends  as  practicable,  and  shall  have  a  length  of      times  the  width  of  the  member. 

122.  Lattice  bars  will  preferably  be  of  soft  steel,  and  be  at  least  2^  inches  wide  for 
f-inch  or  f-inch  rivets,  and  increase  with  the  width  of  the  member.  They  will  make  an  angle 
of  60  degrees  with  the  axis  of  the  member,  and  the  same  angle  with  each  other. 

123.  Double  lattice  will  be  used  in  all  members  having  a  clear  width  between  webs  of 
more  than  20  inches. 

124.  When  necessary,  pin-holes  shall  be  re-enforced  by  plates,  so  that  the  allowed 
pressure  on  the  pins  will  not  be  exceeded.  These  re-enforcing  plates  must  contain  enough 
rivets  to  transfer  proportion  of  the  bearing  pressure,  and  at  least  one  plate  on  each  side 
shall  extend  not  less  than  6  inches  beyond  the  edge  of  the  furthest  batten  plate. 

Pin,  splice,  and  batten  plates  will  be  universal-rolled  plates. 

125.  When  the  ends  of  compression  members  are  forked  to  connect  to  the  pins,  the  ag- 
gregate compressive  strength  of  these  forked  ends  must  equal  the  compressive  strength  of  the 
body  of  the  members  ;  in  order  to  insure  this  result,  the  aggregate  sectional  area  of  the  forked 
ends  at  any  point  between  the  inside  edge  of  the  pin-hole  and  6  inches  beyond  the  edge  of 
the  batten  plate,  shall  be  about  double  that  of  the  body  of  the  member. 

126.  In  compression  chord  sections,  the  material  must  mostly  be  concentrated  at  the 
sides,  in  the  angles  and  vertical  webs. 

127.  Pin-holes  shall  be  bored  exactly  perpendicular  to  a  vertical  plane  passing  through 
Pin-ho\es.       the  centre  line  of  each  member  when  placed  in  a  position  similar  to  that  which  it  is  tooccupy 

in  the  finished  structure. 

The  ends  of  all  members  which  make  contact  joints  shall  be  planed  smooth,  to  eji  .ct 
lengths  and  to  exact  angles,  with  the  axis  of  the  member. 

128.  Abutting  members  must  be  brought  into  close  and  forcible  contact  when  fitted  with 
splice  plates,  and  the  rivet-holes  reamed  in  position  before  leaving  the  works,  all  pieces  being 
match-marked,  so  as  to  fit  in  the  same  position  in  erecting. 


Pin  plates. 


Forked  ends. 


Milled  ends. 


Reamed  splices. 


STRUCTURAL  STEEL  AND  GENERAL  SPECIFICATIONS. 


495 


Pitch  of  rivets  at 
ends. 

Couplers. 


Collision  struts. 


129.  In  all  compression  members  the  pitch  of  rivets  at  the  ends  shall  not  be  over  four 
times  the  diameter  of  the  rivets  for  a  length  equal  to  twice  the  width  of  the  member. 

130.  The  couplers  on  chords  and  end-posts  will  be  at  least  \  inch  thick. 

131.  In  intermediate  posts,  both  web  and  pin  plate  will  extend  beyond  the  pin  for  a  length 
about  equal  to  the  diameter  of  pin-hole. 

132.  Collision  struts  will  be  disposed  to  best  advantage  to  take  a  shock  of  derailed  train 
striking  3^  feet  above  base  of  rail.  They  will  preferably  make  an  angle  of  80  degrees  or  more 
with  the  end-post,  and  may  be  placed  3  feet  away  from  the  point  indicated  above  in  order  to 
increase  the  angle. 

Tension  Members. 


Upset  screw  ends 


Eye-bars. 

133.  All  eye-bars  will  be  solid  forged-steel  bars  of  approved  quality  and  rolled  by  mills 
having  established  reputations  for  the  manufacture  of  eye-bar  steel. 

134.  No  work  on  the  bars  will  be  done  at  a  blue  heat,  and  all  bars  must  be  thoroughly 
annealed  after  forging. 

135.  The  heads  of  eye-bars  shall  be  so  proportioned  that  the  bar  will  break  in  the  body 
instead  of  in  the  eye.  The  form  of  the  head  and  the  mode  of  manufacture  shall  be  subject  to 
the  approval  of  the  Chief  Engineer  of  the  railway  company  before  the  contract  is  made. 

136.  The  bars  must  be  free  from  flaws  and  of  full  thickness  in  the  necks.  They  shall  be 
perfectly  straight  before  boring.  The  holes  shall  be  in  the  centre  of  the  head  and  on  the  cen- 
tre line  of  the  bar. 

137.  The  bars  must  be  bored  of  exact  lengths  and  the  pin-holes  ^'j  inch  larger  than  the 
diameter  of  the  pin. 

All  bars  on  the  same  item  must  be  bored  at  one  setting  of  the  drills  and  at  the  same  tem- 
perature. 

Bars  may  vary  ^  inch  from  ordered  length,  but  bars  on  the  same  item  must  not  vary  -^^ 
inch  in  length. 

Rivet-holes  in  eye-bars  will  be  drilled,  and  have  the  wire  edges  cut  off  the  edges  of  the 
boles. 

138.  Upset  screw  ends  may  be  used  on  steel  bars  if  fully  guaranteed  and  tested  in  full- 
sized  bars.  The  area  at  base  of  thread  and  at  all  parts  of  upset  ends  will  be  15  per  cent  in 
excess  of  the  area  of  the  bar. 


Riveted  Tension  Members. 


Bxcess  at  pia- 
holes. 


139.  Riveted  tension  members  must  be  designed  with  special  care.  Members  with  pin 
connections  will  be  required  to  have  net  areas  across  the  pin-hole,  and  back  of  the  pin-hole, 
respectively  of  150  per  cent  and  80  per  cent  of  the  net  area  required  in  the  body  of  the  mem- 
ber, and  there  will  be  a  corresponding  excess  of  rivets  to  make  this  material  effective. 

The  length  from  back  of  eye  to  end  of  member  must  be  greater  than  the  radius  of  the 

pin. 

140.  In  members  with  riveted  connections  special  care  will  be  used  to  have  the  rivets 
symmetrically  arranged  from  the  centre  line  and  to  avoid  eccentricity  of  stress. 

If  rivets  are  staggered  they  will  all  be  deducted  as  though  they  occurred  at  the  same 
cross-section. 

Rods. 

141.  Rods  for  counters  or  laterals  will,  preferably,  be  either  simple  loops  or  clevis  rods. 

142.  Loop  rods  will  be  of  wrought-iron,  and  must  be  upset  and  welded  in  a  thoroughly 
efficient  and  workmanlike  manner. 

143.  The  eyes  of  all  loop  rods  must  be  bored  to  fit  the  pins. 

Upset  screw  ends  must  have  a  net  section  at  base  of  thread  15  per  cent  greater  than  the 
body  of  the  bar. 

144.  Steel  rods  may  be  used  for  upset  screw  ends,  but  must  be  fully  guaranteed  and  tested 
in  full-sized  section.    They  must  be  annealed  after  forging. 

145.  Clevises,  turnbuckles,  and  sleeve  nuts  must  be  of  approved  pattern,  and  fully  guar- 
anteed. 


49f 


APPENDIX. 


Pins. 

146.  All  main  pins  will  be  made  of  medium  steel. 

The  diameter  of  the  pin  will  not  be  less  than  three-fourths  (|)  the  largest  dimension  ol 
any  tension  member  attached  to  it. 

147.  The  several  members  attached  to  the  pin  shall  be  packed  close  together,  and  all 
vacant  spaces  between  the  chords  and  posts  must  be  filled  with  wrought-iron  filling  rings. 

The  pins  shall  be  turned  straight  and  smooth,  and  shall  fit  the  pin-holes  within  ^  inch. 
They  shall  be  turned  down  to  a  smaller  diameter  at  the  ends  for  the  thread,  and  driven  in 
place  with  a  pilot-nut  when  necessary  to  save  the  thread. 

148.  Nuts  will  be  of  wrought-iron  or  wrought-steel.  There  will  be  a  washer  under  each 
nut,  or  else  Lomas  nuts  will  be  used. 

Lateral  Bracing. 

149.  The  attachment  of  the  lateral  system  to  the  chords  shall  be  thoroughly  efficient.  If 
connected  to  suspended  floor  beams,  the  latter  shall  be  stayed  against  all  motion. 

150.  Preference  will  be  given  to  lateral  bracing  in  the  floor  system,  which  is  capable  of 
resisting  both  compression  and  tension. 

151.  Portals  and  intermediate  knee-braces  shall  be  used  in  all  through  bridges,  so  de- 
signed as  to  form  efficient  and  rigid  connection  between  top  lateral  struts  and  web  members. 

Portals  and  knee-  Where  the  height  of  the  truss  makes  it  practicable,  the  knee-braces  shall  be  replaced  by  suit- 
braces.  °     ,      .  ,      .     ,  .  ,. 

able  cross-section  bracing  at  each  pair  of  intermediate  posts. 

152.  When  brackets  only  can  be  used  for  portal  braces  they  will  have  plate  webs. 
Portals  will  be  as  deep  as  the  head-room  will  allow. 

153.  In  all  deck  bridges  transverse  bracing  shall  be  provided  at  each  panel ;  this  bracing 
shall  be  proportioned  to  resist  the  unequal  loading  of  the  trusses,  and  the  wind  and  centrifugal 

Deck  spans.      forces  ;  the  transverse  bracing  at  the  ends  shall  be  of  the  same  equivalent  strength  as  the  end 
top  lateral  bracing. 

154.  The  hitches  for  lateral  braces  must  be  thoroughly  efficient;  shear  on  field  rivets  will 
be  reckoned  as  per  formula  (I). 

Shoes,  Bed-plates,  etc. 

155.  There  must  be  a  pier  box  or  plate  of  approved  form  under  pedestal  shoes  at  both  ends 
of  sufficient  depth  to  distribute  the  weight  properly  on  masonry.  These  boxes  or  plates  must 
be  at  least  \  inch  thick,  must  have  planed  surfaces,  and  be  of  such  dimensions  that  the  greatest 
pressure  upon  the  masonry  will  not  exceed  250  pounds  per  square  inch,  and  sheet  lead*  not 
less  than  \  inch  thick  shall  be  interposed  between  them  and  the  masonry. 

156.  Where  two  spans  rest  upon  the  same  masonry  a  continuous  plate  not  less  than  finch 
thick  shall  extend  under  the  two  adjacent  bearings. 

157.  All  the  bed-plates  and  bearings  under  fixed  and  roller  ends  must  be  fox-bolted  to  the 
masonry ;  for  trusses,  these  bolts  must  not  be  less  than  \\  inches  diameter ;  for  plate  and  other 
girders,  not  less  than  \  inch  diameter.  The  contractor  must  furnish  all  bolts,  drill  all  holes, 
and  set  bolts  to  place  with  sulphur. 

158.  All  bridges  over  75  feet  span  shall  have  at  one  end  nests  of  turned  friction  rollers, 
formed  of  wrought  steel,  running  between  planed  surfaces.  The  rollers  shall  not  be  less  than 
2|  inches  diameter,  and  shall  be  so  proportioned  that  the  pressure  per  lineal  inch  of  roller  shall 
not  exceed  the  product  of  the  square  root  of  the  diameter  of  the  roller  in  inches  multiplied  by 
500  pound  (500  ^ d).\  Bridges  less  than  75  feet  span  will  be  secured  at  one  end  to  the  ma- 
sonry, and  the  other  end  shall  be  free  to  move  by  sliding  upon  planed  surfaces. 

159.  Friction  rollers  must  be  so  arranged  as  to  be  readily  cleaned  and  to  retain  no  water. 

160.  While  the  roller  ends  of  all  trusses  must  be  free  to  move  longitudinally  under  change 
of  temperature,  they  shall  be  anchored  against  lifting  or  moving  sideways. 

Workmanship  on  Medium  Steel. 

161.  Medium  steel  will  be  subject  to  the  general  conditions  given  under  "Workmanship" 
above,  and  in  addition  to  the  following  requirements: 


Two  spans. 


Bolts. 


Rollers. 


So  lateral  move- 
ment. 


[*  Sheet  aluminum  is  now  used  extensively  for  this  purpose.— J.  B.  J.] 
f  See  Arts.  254  and  255. 


STRUCTURAL  STEEL  AND  GENERAL  SPECIFICATIONS. 


497 


(d)  All  sheared  and  hot-cut  edges  shall  have  not  less  than  \  inch  of  metal  removed  by 
planing.    (Lattice  bars  only  will  be  exempted  from  this.) 

(S)  All  punched  holes,  will  be  reamed  to  a  diameter  \  inch  larger,  so  as  to  remove  all  the 
sheared  surface  of  the  metal. 

{c)  No  sharp  or  unfilleted  re-entrant  corners  will  be  allowed. 

{d)  All  rivets  will  be  steel. 

{e)  Any  piece  which  has  been  partially  heated  or  bent  cold  will  be  afterward  wholly  an- 
nealed. 

Quality  of  Material.  * 
Wrought  Iron. 

162.  All  wrought-iron  must  be  tough,  ductile,  fibrous,  and  of  uniform  quality  for  each  class, 
straight,  smooth,  free  from  cinder-pockets,  flaws,  buckles,  blisters,  and  injurious  cracks  along 
the  edges,  and  must  have  a  workmanlike  finish.  No  specific  process  or  provision  of  manufac- 
ture will  be  demanded,  provided  the  material  fulfils  the  requirements  of  these  specifications. 

163.  The  tensile  strength,  limit  of  elasticity,  and  ductility  shall  be  determined  from  a  stan- 
dard test  piece  not  less  than  \  inch  thick,  cut  from  the  full-sized  bar,  and  planed  or  turned 
parallel.  The  area  of  cross-section  shall  not  be  less  than  \  square  inch.  The  elongation  shall 
be  measured  after  breaking  on  an  original  length  of  8  inches. 

164.  The  tests  shall  show  not  less  than  the  following  results: 


Ultimate  Strength. 

Limit  of  Elasticity. 

Elongation  in  8  inches. 

Pounds  per  square  inch. 

Pounds  per  sq.  inch. 

Per  cent. 

50,000 

26,000 

18 

48,000 

26,000 

15 

48,000 

26,000 

12 

"      "    over    "     "  "   

46,000 

25,000 

10 

165.  When  full-sized  tension  members  are  tested  to  prove  the  strength  of  their  connec- 
tions, a  reduction  in  their  ultimate  strength  of  (500  x  width  of  bar)  pounds  per  square  inch  will 
be  allowed. 

166.  All  iron  shall  bend,  cold,  180  degrees  around  a  curve  whose  diameter  is  twice  the 
thickness  of  piece  for  bar  iron,  and  three  times  the  thickness  for  plates  and  shapes. 

167.  Iron  which  is  to  be  worked  hot  in  the  manufacture  must  be  capable  of  bending 
sharply  to  a  right  angle  at  a  working  heat  without  sign  of  fracture. 

168  Specimens  of  tensile  iron  upon  being  nicked  on  one  side  and  bent  shall  show  a  frac- 
ture nearly  all  fibrous. 

169.  All  rivet  iron  must  be  tough  and  soft,  and  be  capable  of  bending  cold  until  the  sides 
are  in  close  contact  without  sign  of  fracture  on  the  convex  side  of  the  curve. 

170.  Samples  from  each  rolling  will  be  tested,  and  also  widely  differing  gauges  of  the  same 
section. 


Steel. 

171.  All  steel  will  be  uniform  in  quality,  low  in  phosphorus,  and  from  works  of  estab- 
lished reputation. 

172.  The  phosphorus  in  all  melts  of  acid  open-hearth  steel  must  be  less  than  o.  10  per 
Phosphorus.      cent,  and  in  all  Bessemer  or  basic  steel  must  be  less  than  0.08  per  cent.    Certified  analyses 

of  all  melts  will  be  furnished  the  Engineer  free  of  charge. 

173.  The  material  will  be  tested  in  specimens  of  at  least  one-half  square  inch  section,  cut 
^"'^maTriai"'**'*''  ^^^'^       finished  material.    Each  melt  of  steel  will  be  tested,  and  each  section  rolled,  and 

also  widely  differing  gauges  of  the  same  section. 

*  See  also  Specifications  for  Structural  Steel  for  Modern  Railroad  Bridges  (1894),  by  Geo.  H.  Thomson,  Consulting 
Engineer,  Grand  Central  Station,  New  York  ;  also  Specifications  for  Structural  Steel,  by  H.  H.  Campbell  Trans  Am 
See.  C.  E.,  Vol.  XXXIII  (1895). 


* 


498  APPENDIX. 

174.  If  several  different  sections  are  rolled  from  the  same  melt  of  steel,  all  of  them  may 
be  tested  at  the  discretion  of  the  inspector. 
Melt  numbers.  '75-  All  finished  material  will  be  plainly  and  distinctly  marked  with  correct  melt 

numbers. 

176.  If  the  melt  number  is  lacking,  or  is  illegible,  or  has  been  changed,  the  material  may 
be  condemned  at  the  discretion  of  the  inspector. 

177.  If  first  tests  are  unsatisfactory,  the  material  will  not  be  accepted  unless— 
Re-tests.             (i)  A  majority  of  the  tests  fill  the  specifications. 

(2)  All  the  tests  show  good  material  of  reasonable  uniformity. 


Soft  Steel. 

178.  Soft  steel  may  be  used  under  the  same  conditions  as  wrought-iron  except — 

(1)  It  must  be  used  consistently.  The  occasional  use  of  pieces  of  steel  will  not  be  per- 
mitted. 

(2)  It  must  not  be  welded. 

(3)  The  thickness  of  material  subjected  to  punching  will  be  limited  to  \  inch,  excepting 
{a)  in  plate  girders  over  50  ft.  long,  in  which  it  may  be  inch  ;  (b)  in  top  chords  and  end- 
posts,  in  which  it  may  be  \  inch  ;  and  (tj  in  shoes,  pedestals,  and  bed-plates,  m  which  it  may 
be  f  inch  thick. 

179.  Soft  steel  when  tested  as  described  above  must  meet  the  following  requirements: 
An  elastic  limii  of  at  least  32,000  lbs.  per  square  inch. 

An  ultimate  strength  of  54,000  to  62,000  lbs.  per  square  inch. 
An  elongation  in  8  inches  of  at  least  25  per  cent. 
A  reduction  of  area  of  at  least  45  per  cent. 

For  web  plates  over  36  inches  wide  the  elongation  will  be  reduced  to  20  per  cent  and  the 
Web  plates.     reduction  of  area  to  40  per  cent. 

180.  It  must  bend  cold  180  degrees  and  close  down  on  itself  without  cracking  on  the  out- 
side. 

181.  When  |-inch  holes  pitched  J  inch  from  a  roll-finished  or  machined  edge  and  2 
inclies  between  centres  are  punched  the  metal  must  not  crack  ;  and  when  |-inch  holes  pitched 

inch  between  centres  and      inches  from  the  edge  are  punched,  the  metal  between  the  holes 
must  not  split. 
Rivets.  AH  rivets  will  be  soft  steel. 

Medium  Steel. 

182.  Medium  steel  only  will  be  used  for  eye-bars  and  main  pins,  and  it  may  be  used  for 
other  members  under  conditions  given  in  §  161. 

183.  Medium  steel  when  tested  as  described  above  must  meet  the  following  require- 
ments : 

An  elastic  limit  of  at  least  35,000  lbs.  per  square  inch. 

An  ultimate  strength  of  from  60,000  to  70,000  lbs.  per  square  inch. 

An  elongation  in  8  inches  of  at  least  20  per  cent. 

A  reduction  of  area  of  at  least  40  per  cent. 

It  must  bend  180  degrees  011  itself  around  a  ij-inch  round. 

184.  Full-sized  eye  bars,  when  tested  to  destruction,  must  show  an  ultimate  strength  of 
Eye-bar  tests,    at  least  56,000  pounds,  and  stretch  at  least  10  per  cent  in  a  gauged  length  of  10  feet. 

There  will  be  two  bars  tested  from  each  span. 


Inspection. 

185.  Ample  facilities  for  inspection  and  testing  of  material  and  workmanship  must  be 
furnished  to  the  Chief  Engineer  of  the  railway  company  or  his  assistants.  Tests  on  small 
specimens  to  determine  the  quality  of  materials  will  be  made  free  of  charge  to  the  railway 
company  before  any  work  is  done  on  the  material. 

186.  Full-sized  parts  of  the  structure  may  be  tested  at  the  option  of  the  Engineer  of  the 
railway  company,  and  shall  be  paid  for  at  cost  less  their  scrap  value,  if  they  prove  satisfactory. 
If  the  test  is  not  satisfactory  the  contractors  will  receive  no  recompense. 


STRUCTURAL  STEEL  AND  GENERAL  SPECIFICATIONS. 


499 


Painting. 

1S7.  All  surfaces  in  contact  with  each  other  must  receive  one  coat  of  red  oxide-of-iron 
paint  or  other  metallic  paint  mixed  in  pure  linseed  oil,  approved  on  sample  by  the  Engineer 
of  the  railway  company. 

188.  All  work  before  leaving  the  shops  must  be  thoroughly  cleansed  from  all  loose  scale 
and  rust,  and  be  given  one  good  coat  of  pure  raw  linseed  oil,  well  worked  into  all  joints  and 
open  spaces. 

189.  All  surfaces  that  will  be  inaccessible  after  erection  must  receive  one  coat  of  the 
approved  paint  during  erection,  the  iron  to  be  perfectly  cleaned  before  painting. 

190.  The  railway  company  will  paint  the  structure  after  erection. 

191.  All  planed  or  turned  surfaces  must  be  coated  with  white  lead,  mixed  with  tallow, 
before  shipment. 

192.  No  painting  will  be  allowed  during  wet  or  freezing  weather. 

False-work. 

193.  The  contractor  will  furnish  all  necessary  false-work  and  remove  the  same  after  the 
completion  of  the  bridges,  leaving  the  several  streams  and  rivers  unobstructed,  except  the 
actual  space  occupied  by  the  masonry. 

Risks. 

194.  The  contractor  shall  assume  all  risks  from  floods  and  storms,  and  also  casualties  of 
every  description,  and  must  furnish  all  material  and  labor  incidental  to  or  in  any  way 
connected  with  the  manufacture,  erection,  and  maintenance  of  the  structure  until  its  final 
acceptance. 

195.  If  any  patented  parts  be  used,  the  contractor  shall  protect  the  railway  company 
against  any  and  all  claims  on  account  of  such  patents. 

Preservation  of  Old  Material. 

196.  In  the  erection  of  bridges,  the  contractor  will  be  required  to  remove  all  old  material 
in  such  a  manner  as  will  not  impair  its  future  use  in  structures  similar  to  that  from  which  it 
was  taken,  unless  otherwise  agreed. 

Maintaining  Tracks. 

197.  The  contractor  will  be  required  to  maintain  the  track  in  proper  condition  for  the 
passage  of  all  schedule  trains  without  delay,  except  when  previously  arranged  for  by  the 
Division  Superintendent. 

Final  Test. 

198.  Before  the  acceptance  the  Chief  Engineer  of  the  company  may  make  a  thorough 
test  by  passing  over  each  structure  the  specified  trains  or  their  equivalent  at  a  speed  not  ex- 
ceeding thirty  miles  an  hour,  and  bringing  them  to  a  stop  at  any  point  by  means  of  the  air  or 
Other  brakes,  or  by  resting  the  maximum  train  load  upon  the  structure  for  such  period  of  time 
as  he  may  deem  proper. 

Specifications  for  Second-class  Bridge  Superstructure. 

For  Divisions  and  Bratiches  Carrying  Light  Traffic. 

I.  The  loading  will  be  the  same  as  for  first-class  bridges,  and  the  bridges  fully  conform 
to  the  specifications  for  first-class  bridge  superstructures,  excepting  as  follows; 

(i)  Outside  of  the  lateral  system,  tlie  stresses  due  to  wind  and  centrifugal  force  need  not 
be  provided  for  unless  they  exceed  40  per  cent  of  the  stresses  due  to  dead  and  live  load. 


500  APPENDIX. 

(2)  The  unit  stresses  will  be  modified  as  follows  : 

TENSION. 


Wrought  Iron. 

Soft  Steel. 

Medium  Steel. 

8,500 

8,500 

Will  not  be  used 

COMPRESSION. 

Wrought  Iron. 

Soft  Steel. 

Medium  Steel. 

/    ,  min.  \  I 

7,000  llH  1  —  30  — 

\    '  max.'  r 

I    ,  min.\  / 

7,000  li  H  1  —  40  — 

\       max./  r 

10^  greater  than  iron 

<<          ti           <<  <c 

20%  greater  than  iron 

««                H             (f  t€ 

Stresses  not  specified  will  be  the  same  as  for  first-class  bridge  superstructures. 

(3)  One  sixth  (|)  of  webs  of  girders  will  be  considered  available  section  in  each  flange, 
except  at  web  splices,  where  the  full  section  shall  be  provided  for  by  extending  the  flange 
plate  or  by  the  addition  of  cover-plates. 


MANUFACTURE  AND  INSPECTION  OF  IRON  AND  STEEL  STRUCTURES.  501 


APPENDIX  B. 

PROCESSES  IN  THE  MANUFACTURE  AND  IN  THE  INSPECTION  OF  IRON  AND  STEEL 

STRUCTURES.  * 

Of  the  four  natural  divisions  into  which  the  construction  of  a  metallic  framed  structure  is  divided,  i.e., 
Design,  Mill-work,  Shop-work,  and  Erection,  the  first  has  been  treated  in  the  body  of  this  work,  and  of  the 
remainder,  the  mill-work  and  shop-work  alone  will  be  considered  here. 

MILL-WORK. 

Processes  in  Manufacture. — Iron. — The  order  of  the  manufacturing  processes  are  as  follows: 

1.  Making  the  Pig. 

2.  Puddling. 

3.  Rolling  the  Muckbars. 
.               4.  Cutting  and  Piling. 

Mill-work  m  Iron  ^  5.  Railing. 

6.  Straightening. 

7.  Working  and  Shearing. 

8.  Inspection. 

For  what  is  called  double-refined  iron,  after  item  No.  5,  it  would  be  necessary  to  insert  two  more  items,  i.e., 

Cutting  and  Piling,  and  Rerolling. 

The  Pig  is  made  by  melting  the  ore  with  a  flux,  in  a  blast-furnace,  the  product  being  run  out  into  a 
series  of  small  "  pens  "  to  cool. 

This  is  then  puddled  in  a  square  box  furnace,  also  called  a  reverberatory  furnace,  where  it  is  remelted, 
much  of  the  combined  carbon  and  other  impurities  burned  out  of  it,  when  the  iron  is  gradually  gathered 
by  the  puddler  into  spongy  masses  called  puddle-balls,  weighing  about  seventy-five  pounds  each.  These  are 
then  removed  from  the  furnace  and  taken  to  either  a  hammer  or  a  cam-squeezer,  and  worked  down  into  a 
shape  suitable  for  rolling  into  muck-bar.  This  process  also  removes  most  of  the  slag  contained  in  the  ball. 
The  material  is  next  sent  through  the  muck-rolls  and  reduced  to  muck-bar.  These  bars  are  usually  about 
four  or  five  inches  wide  and  one  inch  thick,  and  are  very  rough  in  appearance. 

This  muck-bar,  together  with  waste  scrap-iron,  is  then  cut  up  into  lengths,  usually  not  over  six  or  seven 
feet,  depending  upon  the  size  of  the  piece  to  be  rolled,  and  reheated  in  an  oven  and  then  passed 
through  the  rolls.  As  has  been  said,  for  double-refined  iron  this  material  would  again  be  cut  up,  piled,  re- 
neated,  and  again  rolled  into  flat  bars,  and  these  replied  and  rolled  to  the  final  shapes.  This  material  is 
much  stronger  and  more  homogeneous  than  the  first  product,  which  is  called  Refined  Iron  or  Merchant  Bar. 
Alter  the  material  has  been  rolled  to  its  final  forms  it  is  run  out  on  a  series  of  skids  called  the  hot-bed, 
wnere  it  is  allowed  to  cool.  From  here  it  goes  through  the  straightening  machine.  This  may  be  either  a 
gag-press  or  a  train  of  rolls,  three  below  and  two  above.  The  latter  is  much  the  better,  producing  straighter 
bars  with  less  injury  to  the  material. 

After  coming  from  the  straightening  machine  the  material  is  marked  and  sheared  to  length,  and  then 
inspected.  Each  piece  is  marked  in  white-lead  with  its  true  dimensions,  and  also,  in  the  case  of  steel,  with 
its  heat  and  bloom  number.    The  material  is  now  ready  to  be  shipped  to  the  bridge  manufacturer. 

Steel-work. — Steel  is  made  by  various  processes,  of  which  the  best,  to  date,  are  the  Open-hearth  and 
the  Bessemer.  For  a  complete  discussion  of  these  processes  see  standard  works  on  the  metallurgy  of  iron 
and  steel.t  The  Open-hearth  processes  (for  there  are  several),  though  slower  (requiring  from  seven  to  ten  hours 
for  one  heat,  while  the  Bessemer  blow  can  be  made  in  half  an  hour),  are  considered  by  many  engineers  to  be 
more  thorough,  producing  a  more  homogeneous  material  than  the  Bessemer  product.  With  the  same  grade 
of  ore  in  each  case  this  is  undoubtedly  true.  A  cheaper  grade  of  ore  is  employed  usually  in  the 
Open-hearth  process.  When  the  melt  is  finished  it  is  run  oft'  into  ingot-moulds.  The  ingots  are  about 
eighteen  inches  square  at  one  end  and  twenty  inches  square  at  the  other,  and  about  six  feet  long.  One 

*  See  also  Specifications  for  Structural  Steel  for  Modern  Railroad  Bridges,  and  Instructions  to  Inspectors,  by  Geo. 
H.  Thomson,  Consulting  Engineer,  Grand  Central  Station,  New  York. 

f  Especially  Howe's  Metallurgy  of  Steel ;  also  a  paper  on  Specifications  for  Structural  Steel,  by  H.  H.  Campbell,  in 
Trans,  Am.  Soc.  C,  £.,  Vol.  XXXIII. 


S02 


APPENDIX. 


blow  from  the  Bessemer  converter  makes  three  such  ingots,  while  a  meit  or  neat  irom  tne  open-hearth 
furnace  will  make  six. 

From  each  heat  or  blow  there  is  also  cast  a  small  billet  about  four  mches  square,  which  is  rolled  down 
into  a  three-quarter-inch  round,  from  which  the  heat  or  blow-tests  are  made  lor  determining  me  quality  of 
the  steel.  A  chemical  analysis  is  also  made  of  each  heat.  The  large  ingots  are  next  reheated,  and  taken 
to  the  blooming-mill,  where  they  are  rolled  down  and  cut  into  bloorns.  The  size  of  these  will  vary  according 
to  the  order  in  hand.    Each  bloom  is  supposed  to  make  a  certain  number  of  ordered  pieces. 

The  remaining  processes  in  the  mill  are  the  same  for  steel  as  tor  iron,  with  the  exceotion  noted. 

Inspection. — In  the  inspection  of  iron,  no  tests  can  be  made  before  the  material  is  rolled.  Specimens 
are  then  selected,  two  or  three  or  more,  depending  upon  the  size  of  the  order,  from  each  size  and  shape,  and 
from  these  test  specimens  are  cut,  about  i8"  in  length,  i"  in  v/idth  in  the  reduced  portion,  the  thickness 
being  left  the  same  as  that  from  which  the  specimen  was  taken. 

In  iron  each  pile  may  differ  from  all  the  others.  This  difference  is  in  general  very  slight,  the  same 
kind  of  muck-bar  and  the  same  kind  of  scrap  being  used.  Nevertheless,  for  this  reason  more  tests  are  made 
of  iron  than  of  steel. 

In  steel,  tests  are  made  on  the  |"  round  specimens  before  the  material  is  rolled.  These  tests  wiU  usually 
run  a  little  below  the  final  finished  material  tests  in  elastic  limit  and  ultimate  strength,  and  a  little  above 
them  in  elongation  and  reduction.  Allowance  should  be  made  for  this  variation  in  the  acceptance  of  the 
heat. 

After  the  material  is  rolled  test  specimens  are  taken,  one  from  each  size  of  each  heat  for  tension  tests. 
Specifications  usually  require  also  bending,  nicking  and  bending,  punching,  and  tempering  tests.  One  test 
of  each  kind  for  each  heat  is  all  that  would  be  required. 

These  represent  the  physical  tests  of  the  material.  For  each  of  them  the  inspector  can  get  numerical 
values.  For  the  superficial  inspection,  however,  he  cannot  do  this,  and  here  he  is  largely  left  to  his  own 
judgment.  Rejections  at  this  point  are  for  various  reasons,  such  as  unwelded  or  ragged  edges,  burns,  cinder 
spots,  blisters,  etc.,  not  easily  described,  but  learned  by  experience. 

In  inspecting  the  material  the  inspector  should  have  a  copy  of  the  mill  order  and  check  off  such  as  he 
accepts,  so  that  he  as  well  as  the  mill  people  may  know  how  much  remains  to  be  furnished. 

With  his  hammer  he  should  also  stamp  every  accepted  piece,  putting  a  ring  of  white-lead  around  the 
mark.  This  will  save  much  trouble  at  the  shop,  and  should  never  be  neglected.  When  the  material  is 
shipped  he  should  compare  the  invoices  with  his  mill  order,  and  see  that  no  more  thnn  he  has  accepted  has 
been  shipped.  In  iron,  superficial  inspection  of  the  muck-bar  is  sometimes  required,  and  also  of  the  piling. 
If,  however,  the  finished  material  is  to  be  thoroughly  tested  these  restrictions  are  not  necessary. 

It  is  not  just  to  demand  a  certain  grade  of  article  and  also  specify  how  it  shall  be  obtained.  The  same 
is  true  with  regard  to  the  limitations  placed  upon  the  chemical  composition  of  steel.  The  physical  tests 
should  be  sufficient  to  develop  its  capacity  for  performing  the  work  it  has  to  do. 

SHOP-WORK. 

Processes  in  Manufacture. — The  various  processes  in  the  shop  are  as  follows: 

1.  Straightening  (when  necessary). 

2.  Marking-off  and  punching. 

3.  Straightening. 

4.  Reaming. 

5.  Assembling. 

6.  Reaming. 

7.  Riveting. 

8.  Facing. 

9.  Boring. 

10.  Finishing. 

1 1.  Fitting  up. 

12.  Inspecting. 

13.  Oiling  and  Painting. 

14.  Shipping. 

This  table  applies  to  large  members  as  a  whole,  such  as  posts,  chords,  girders,  etc.  On  any  given  con- 
tract there  will  be  more  or  less  machine  and  blacksmith  shop-work  not  provided  for  in  the  table. 

Straightening. — This  is  often  necessary,  either  because  it  has  not  been  done  properly  at  the  mill  or 
has  been  abused  in  handling.    The  work  cannot  be  laid  off  well  if  the  material  is  very  crooked. 


Order  of  processes  in  shop : 


MANUFACTURE  AND  INSPECTION  OF  IRON  AND  STEEL  STRUCTURES.  503 

Marking-OFF  and  Punching. — Templets  are  made  for  the  proper  gauge  and  pitch  of  rivets,  and  from 
these  the  material  is  marked  off.  Upon  this  work  single  punches  are  used.  For  complementary  chord 
angles  no  templets  are  necessary.  They  are  clamped  together  and  put  through  a  rack  punch.  Multiple 
punches  are  used  on  web  plates.  With  these  the  entire  row  of  stiffener  rivet-holes  can  be  punched  at  one 
setting.  The  rack  and  multiple  punches  save  a  great  deal  of  time,  but  they  require  more  skill  and  care  to 
achieve  as  good  results  as  are  obtained  from  the  templet,  single-punch  work.  A  broken  gauge-line  on  choic' 
angles  makes  very  unsightly  work. 

All  material  is  stretched  slightly  by  the  process  of  punching.  This  is  very  noticeable  in  chord  angles, 
especially  if  both  legs  are  punched,  in  which  case  it  may  amount  to  as  much  as  J  to  i  inch  in  20  feet.  It  will 
vary  according  to  the  size  of  the  angle  and  size  and  pitch  of  the  rivets.  The  effect  upon  web  plates  is  to 
stretch  a  small  strip  of  the  plate  next  to  the  punched  edge.  This  results  in  a  continuous  series  of  short 
buckles  along  this  edge,  impossible  to  take  out  in  bolting  up  the  member. 

Straightening. — On  account  of  this  buckling  of  the  material  in  punching,  the  material  should  always 
be  sent  to  the  rolls  and  straightened  before  assembling.  This  is  a  very  important  item,  but  one  which  is 
sometimes  entirely  disregarded.  If  it  is  not  done,  the  angles  and  web  plate  cannot  be  made  to  come 
together.  Then  in  the  ordinary  process  of  rapid  riveting,  the  rivet  will  be  sufficiently  hot  after  the  pressure 
is  removed  to  allow  the  spring  between  the  angles  and  plate  to  distort  or  draw  the  rivet  slightly.  When  the 
next  rivet  goes  in  it  draws  the  material  up  again,  and  this  leaves  the  first  rivet,  if  not  loose,  at  any  rate  not 
cupped  down.  This  will  be  true  of  every  rivet  driven  unless  the  pressure  is  held  on  an  unusual  length  of 
time.  In  cases  like  this  it  is  not  unusual  for  as  high  as  20  per  cent  of  the  rivets  in  a  girder  to  be  cut  out. 
The  finished  member  also  never  looks  as  well  as  if  the  material  had  been  straightened. 

Reaming.— This  may  be  done  by  hand  or  by  flexible  tube-reamers  attached  to  the  shafting.  The  latter 
method  is  of  course  much  the  better.  Nearly  all  specifications  require  that  the  material  shall  be  punched 
to  a  diameter  less  and  reamed  to  a  diameter      in.  greater  than  that  of  the  rivet.    Some  specifications 

require  the  punched  holes  to  be  reamed  to  a  diameter  \  in.  larger  than  the  rivet.  This  is  for  the  purpose  of 
removing  the  material  affected  by  the  punch.  It  has  not  been  established,  however,  that  in  soft  steel  the 
material  around  the  hole  has  been  materially  injured  in  punching  plates  less  than  |  in.  in  thickness.* 

On  the  punch  side  of  the  plate  the  material  remains  in  its  normal  condition  and  uninjured,  but  on  the 
die  side  the  effect  is  different  and  the  material  is  injured  from  the  cold  flowing  produced  here.  This  injury 
will  vary  directly  with  the  thickness  of  the  plate,  and  with  the  bluiitness  of  the  edges  of  punch  and  die. 

Assembling. — The  next  process  is  to  collect  all  the  various  pieces  forming  a  member  of  the  structure 
and  bolt  them  up  preparatory  to  riveting. 

Reaming. — Before  going  to  the  riveter,  however,  a  reamer  is  passed  through  all  the  holes  to  make  sure 
that  the  rivet  will  enter.  It  is  thus  seen  that  we  have  two  reamings.  Owing  to  the  stretching  of  the 
material  in  the  punching,  the  angles  stretching  more  than  the  web,  and  to  accidental  causes,  it  is  impossible 
to  make  all  the  holes  fit.  In  a  30- ft.  girder  the  holes  near  the  ends  may  not  match  by  i\  in.  The  reamer 
is  put  through  and  takes  off  not  more  than  ^  in.  from  the  angles  and  the  remaining  ^  in.  from  the  web. 
This  leaves  a  pocket  \  in.  deep  at  the  web  into  which  the  rivet  must  back  up  in  order  to  fill  the  hole  and  be 
able  to  transmit  stress.  It  takes  a  good  riveting-machine  to  do  this  well.  Usually  it  would  not  be  done. 
For  this  reason  the  writer  is  in  favor  of  punching  the  material  to  a  diameter  \  in.  less  than  the  diameter  of 
the  rivet,  and  of  having  but  one  reaming,  that  being  done  after  the  pieces  arc  assembled.  Some  would 
make  the  objection  to  this  method,  that  all  of  the  injured  material  around  the  hole  would  not  be  removed. 
But  as  it  would  be  better  to  have  the  vacant  space  filled  with  anything  than  with  nothing  at  all,  and  as 
what  proof  there  is  available  upon  the  subject  seems  to  indicate  that  no  real  injury  has  been  done  to  the 
working  strength  of  the  member,  the  method  would  seem  to  be  worthy  of  adoption. 

Riveting. — Whenever  it  is  practicable,  rivets  are  driven  by  machine  riveters.  These  are  of  many 
kinds.  Compressed  air,  steam,  and  water  furnish  three  different  kinds  of  power,  and  each  of  these  are 
applied  to  several  different  styles  of  machines.  They  may  be  divided  into  two  general  classes,  direct  and 
indirect  acting.  By  a  direct-acting  riveting-machine  is  meant  one  in  which  the  ram  moves  in  the  line  of 
final  pressure  throughout  the  stroke.  The  indirect-acting  machines,  instead  of  a  ram  acting  directly  from 
the  piston,  have  two  jaws  pivoted  in  the  middle,  the  power  being  applied  at  one  end  and  the  pressure  on  the 
rivet  at  the  other.  The  jaw  therefore  moves  in  the  arc  of  a  circle.  These  machines  act  well  enough  where 
the  thickness  of  material  through  which  the  riveting  is  being  done  is  equal  to  the  distance  between  the  cups 
on  the  fixed  and  movable  jaws  when  the  faces  of  these  cups  are  parallel.  If  these  distances  are  not  equal, 
lop-sided  rivets  will  be  the  result.  To  make  good  rivet-heads  with  such  a  machine,  therefore,  requires  for 
each  diameter  of  rivet  a  set  of  cups,  one  for  each  length  of  rivet.  Some  of  these  cups  may  be  dispensed 
with  by  having  a  set  of  washers  to  go  under  the  cup. 


*  See  Appendix  A  on  this  subject. 


504  APPENDIX. 

The  indirect-acting  riveter  is  not  as  satisfactory  as  the  direct-acting,  and  is  very  often  forbidden  on 
large  contracts. 

The  hydraulic,  direct-acting  machine  is  the  most  satisfactory  of  all.  Steam  and  compressed  air  work 
well  on  the  average,  but  when  it  happens  that  all  the  machines  in  a  shop  are  working  to  their  full  capacity 
at  the  same  lime,  the  pressure  usually  runs  low. 

Facing. — All  abutting  ends  of  members  need  to  be  planed  ofT.  This  is  done  by  means  of  a  rotary 
planer  or  facer.  This  machine  has  a  face-plate  revolving  in  a  vertical  plane,  into  and  at  right  angles  to 
which  are  set  a  series  of  cutting  tools,  from  three  to  six  inches  apart,  forming  a  circle  around  the  centre  of 
the  face.  The  piece  to  be  planed  is  bolted  to  a  bed  in  front  of  this  face,  and  the  latter  is,  by  means  of  a  screw, 
fed  across  the  end  of  the  piece,  each  tool  taking  off  a  slight  chip.  The  frame  holding  the  face-plate  can 
also  be  revolved  around  a  vertical  axis  any  required  number  of  degrees  so  as  to  plane  the  end  on  a  mitre 
if  so  desired,  as,  for  example,  the  ends  of  batter-posts. 

Boring. — For  boring  pin-holes,  etc.,  two  kinds  of  machines  are  used,  vertical  and  horizontal.  The 
latter  is  generally  preferable,  allowing  a  more  accurate  adjustment  of  the  member. 

Finishing. — This  includes  all  liand  work  necessary  to  finish  the  piece,  such  as  putting  on  of  brackets, 
driving  such  rivets  as  could  not  be  driven  by  machine,  chipping,  and  making  whatever  changes  the  inspector 
may  require. 

Oiling  and  Painting. — Before  shipping,  the  material  should  be  cleansed  of  loose  scale  and  rust,  and 
oiled  with  a  coat  of  pure  raw  linseed  oil.  After  this  it  should  be  painted  with  some  good  quality  of  paint, 
but  this  is  not  usually  done  at  the  shop. 

Inspection. — On  a  large  contract  of  from  5000  to  10,000  tons  of  material,  the  best  shop  inspection  is 
secured  by  the  employment  of  special  inspectors  who  give  the  work  their  personal  supervision  and  report 
directly  to  the  Chief  Engineer.  The  mill  work,  however,  is  more  economically  done  by  the  agents  of  some 
regular  Inspection  Bureau. 

Such  a  contract  as  the  one  named  above  requires,  in  the  shops,  two  men,  one  being  the  Chief  Inspector 
and  the  other  his  Assistant.  The  Chief  Inspector  does  all  of  the  actual  inspection,  receives  reports  from 
mill  inspectors,  looks  out  for  possible  causes  of  delay,  and  has  the  general  supervision  of  all  the  work.  He 
reports  weekly  to  the  Chief  Engineer.  His  assistant  helps  in  taking  measurements,  keeps  up  invoice 
reports  (checking  weights  of  same  if  the  contract  is  by  the  pound),  keeps  track  of  material  for  the  daily 
progress  report,  makes  tests  of  material,  and  does  many  other  things  of  a  similar  nature. 

The  Records. — If  there  is  a  time-penalty  clause  in  the  specifications,  as  there  usually  is,  another  object 
of  the  mspection  is  to  be  able  to  testify  as  to  the  cause  and  character  of  any  delay  in  the  prosecution  of  the 
work.  To  keep  account  of  the  material  in  such  a  way  that  this  can  be  done  is  not  a  simple  matter.  On 
such  a  contract  as  has  been  named  it  would  require  the  inspector  to  know,  at  any  time,  the  exact  condition 
of  from  12,000  to  24,000  pieces  of  material.  This  is  not  as  difficult  a  task  as  it  would  seem  at  first. 
Of  these  12,000  to  24,000  pieces  not  all  are  changing  their  condition  each  day.  Probably  half  of  them  are. 
Then,  too,  they  go  in  groups,  so  that  we  can  greatly  reduce  the  number. 

To  keep  this  record  requires  a  day-book  ruled  as  below  : 


Le/t-kand  fiage.  Right-hnnd  page. 


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General  Remarks. 

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In  these  columns,  each  day,  the  inspector  enters  the  designation  of  the  piece  which  has  received  that 
particular  treatment.  It  will  not  always  be  possible  for  him  to  get  these  independently  of  the  contractor, 
but  that  is  not  necessary.  If  he  goes  about  it  in  the  right  way  the  contractor  will  allow  him  access  to  the 
daily  reports  of  the  shop,  which  of  necessity  are  as  correct  as  maybe.  From  these  he  can  readily  get  the  last 
five  columns  and  enter  them  directly  into  his  day-book.  Punching  and  reaming  are  more  difficult  to  get. 
But  these  also  in  most  shops  he  can  get  from  the  shop  records.  But  they  will  probably  give  the  record  for 
the  component  pieces  and  not  for  the  whole  member.    In  this  case  he  cannot  use  the  day-book  at  first,  but 


MANUFACTURE  AND  INSPECTION  OF  IRON  AND  STEEL  STRUCTURES.  505 


must  take  the  shop  list  on  which  all  of  the  members  with  their  component  pieces  are  given,  and  a  copy  of 
which  he  can  get. 

On  these  sheets,  using  red  ink  entirely,  he  rules  two  columns,  one  for  punching  and  one  for  reaming,  and 
then  enters  up  these  records  for  each  of  the  essential  component  pieces  with  date.  Then  when  these  have 
all  been  treated  for  that  member  he  enters  it  in  his  day-book  for  the  day  on  which  the  last  really  necessary 
piece  of  the  member  was  finished. 

For  keeping  account  of  the  mill  material  he  checks  of!  on  the  mill  order,  a  copy  of  which  he  has,  the 
pieces  from  invoices  as  they  come  in,  entering  date  of  receipt  of  car.  Thus  the  assistant  has  a  record  of 
every  piece  from  the  time  it  left  the  mill  until  ready  for  finishing  and  inspecting.  The  rest  is  obtained  by 
the  Chief  Inspector  himself. 

When  a  large  number  of  pieces  of  similar  but  slightly-varying  construction  are  under  contract,  the 
inspection  is  sometimes  rendered  difficult  by  the  large  number  of  sheets  of  drawings  involved.  This  diffi- 
culty is  best  obviated  by  the  preparation  of  tables  giving  all  of  the  important  descriptive  features  of  each 
member.    These  tables  are  made  out  in  a  note-book  which  the  Chief  Inspector  always  has  with  him. 

The  table  for  chords  and  posts  is  given  below : 


CHORDS  AND  POSTS. 


No.  of 
Draw- 
ing. 

Name 

of 
Piece. 

Length 
over 
all. 

Length 

be- 
tween 
Pin- 
centres. 

Size  of  Pin-hole. 

Size  of 
Web 
or  Bar. 

Size  of  Chord 
Angles. 

Thick- 
ness of 
Pin- 
bear- 
ing. 

Clearance. 

Cover- 
plates. 

Splice- 
plates. 

Remarks. 

N. 

S. 

Top. 

Bot. 

Inside. 

Out- 
side. 

The  floor-beam  and  stringer  table  would  be  as  follows  : 


FLOOR-BEAMS  AND  STRINGERS. 


No.  of 

Name  of 

Length. 

Size  of  Chord  Angles. 

Size  of  End 

No.  of  Rivets 

in  End 
Connections. 

Size  of  Web. 

Remarks. 

Drawing. 

Piece. 

Top. 

Bottom. 

Stiffeners. 

ft 

In  case  the  floor-beams  have  hangers  or  a  direct  connection  to  the  pin  a  slightly  different  form  would 
be  required. 

In  elevated  or  viaduct  work,  also,  the  stringer  table  would  require  some  extra  columns  to  provide  for 
the  bevels  at  the  ends  on  grades  and  curves,  thus : 


Bevels. 

Vertical. 

Horizontal. 

Fixed  End. 

Expansion  End- 

Fixed  End. 

Expansion  End 

APPENDIX. 


A  very  simple  table  suffices  for  eye-bars.  Other  things  such,  as  pins,  rollers,  bracing  rods,  lateral  plates, 
pedestals,  etc.,  are  all  easily  tabulated.  Often  it  is  neither  necessary  nor  advisable  to  do  this.  They  may  be 
more  easily  inspected  either  from  the  drawings  or  the  shop  lists.  But  this  advantage  of  tabulation  should 
be  taken  into  consideration.  It  enables  the  inspector  to  have  always  with  him,  without  any  inconvenience, 
a  complete  record  of  «// the  pieces.  There  is  no  trusting  to  memory.  If  in  passing  tlirough  the  shop  at 
any  time  he  notices  something  wrong  with  a  certain  piece,  he  enters  it  in  the  column  for  remarks  against 
that  piece,  and  then  notifies  the  foreman  of  that  department.  When  this  piece  comes  up  for  final  inspection 
the  note  is  found  and  that  point  re-examined. 

The  inspector  should  so  arrange  his  work  as  to  inconvenience  the  contractor  as  little  as  possible.  He 
cannot  leave  his  inspection  until  the  material  is  ready  to  ship.  Neither  can  he  inspect  and  accept  the  piece 
in  each  of  its  successive  steps  through  tlie  shop.  The  proper  adjustment  of  the  work  will  vary  somewhat 
for  different  shops  and  for  the  kind  of  work.  But  he  should  always  be  on  hand  when  wanted,  and  should 
keep  a  watchful  eye  on  all  of  the  departments. 

One  objection  to  the  subdivision  of  the  inspection  is  that  the  inspector  finds  it  difficult  to  keep  account 
of  what  has  been  inspected  and  what  has  not.  By  the  method  of  tabulation  spoken  of  above  this  is 
rendered  perfectly  easy  and  reliable.  If  certain  things  on  certain  members  have  been  inspected  before  the 
completion  of  the  member,  some  sign  or  letter  indicating  the  same  is  entered  opposite  each  of  these  mem- 
bers in  his  note-book,  and  the  same  signs  put  upon  the  members  themselves  in  soapstone.  When  the 
member  is  finished  the  remaining  requirements  are  checked. 

Details  of  thk  Work  of  Inspection. — The  inspector  should  see  that  the  material  is  not  so 
crooked  when  it  starts  into  the  shop  as  to  prevent  its  being  properly  laid  off  or  gauged  and  punched. 
After  being  punched  the  material  should  be  straightened. 

The  punch-dies  should  be  examined  occasionally  to  see  that  the  edges  are  sharp  and  unbroken,  and  that 
the  difference  in  diameter  between  the  upper  and  lower  does  not  exceed  -^-^  inch. 

After  being  straightened  and  reamed,  the  various  pieces  forming  a  single  member  are  assembled.  This 
requires  a  good  deal  of  care.  Web  splices  and  all  abutting  sections  should  be  made  to  close  tightly  and  the 
splice-plates  fitted  on  and  reamed  while  in  this  position.  Excessive  drifting,  or  drifting  for  any  purpose 
other  than  bringing  the  pieces  to  the  proper  position,  should  not  be  allowed. 

The  inspector  should  see  that  a  sufficient  number  of  bolts  are  used  to  hold  the  material  snugly  together 
while  being  riveted.  The  inspector  should  see  that  all  stiffeners  fit  tight  against  the  chord  angles,  and  that 
all  surfaces  to  be  riveted  together  are  painted  before  being  bolted  up.  After  being  assembled  all  the  rivet- 
holes  are  to  be  reamed. 

Ordinarily  not  much  trouble  is  experienced  with  power  riveting-machines.  Sometimes,  however,  the 
power  gets  low  and  many  loose  rivets  are  found.  One  cause  for  this  is  the  removal  of  the  bolts  too  far 
ahead  of  the  riveting-machine.  In  testing  rivets  they  should  be  struck  two  short,  sharp  blows,  one  on  each 
side  of  the  head,  with  a  hammer  weighing  about  one  pound,  the  handle  to  which  is  quite  small  in  the  shank, 
allowing  the  absorption,  at  this  point,  of  some  of  the  spring  of  the  hammer.  When  the  handle  is  held  at  the 
proper  point  and  the  rivets  are  solid,  no  jarring  effect  is  felt  in  the  hand.  A  little  experience  enables  one 
to  detect  loose  rivets  by  means  of  the  action  of  this  handle  where  no  rattling  sound  could  be  heard,  and 
where  no  movement  could  be  detected  by  the  finger  placed  at  the  angle  between  rivet-head  and  girder. 

Very  often  there  is  trouble  with  countersunk  rivets  driven  by  a  machine.  The  reason  is  this:  The 
rivets  are  a  trifle  too  long.  This  excess  material  spreads  out  under  the  die  and  overlaps  the  hole.  Being  thin 
this  edge  hardens  quickly,  and  then  no  amount  of  pressure  will  upset  the  body  of  the  rivet  any  further.  It 
will  appear  tight  until  chipped,  when  it  is  often  found  to  be  loose.  Drawings  often  require  flat-head  rivets  in 
certain  places  where  there  is  not  enough  clearance  for  the  hemispherical  head  and  yet  where  all  the  space 
obtained  by  countersinking  is  not  necessary.  On  account  of  the  difficulty  mentioned  above,  such  rivet- 
heads,  less  than  \  inch  in  thickness,  should  not  be  allowed.  If  left  unchipped  it  cannot  be  known  whether 
the  rivet  fills  the  hole  or  not. 

In  hand  riveting,  spring  dollies,  for  holding  up  the  rivet,  should  be  used  where  possible,  especially  for 
heavy  pieces.  These  dollies  consist  of  a  long  bar  of  wood  or  iron  used  as  a  lever,  the  short  end  of  which  is 
bent  up  and  contains  a  cup  that  fits  on  the  head  of  the  rivet.  In  this  way  the  effective  weight  of  the 
dollie,  for  the  resistance  of  the  impact  of  the  sledge,  is  equal  to  whatever  weight  is  used  on  the  long  end, 
multiplied  by  the  ratio  of  the  lever-arms.  With  two  men  on  the  dollie  this  can  be  made  quite  large. 
This  is  assuming  tK.it  the  piece  being  riveted  is  sufficiently  heavy  to  hold  them  up. 

Spring  dollies  cannot  be  used  on  light  work.  For  this,  simple  hand  dollies,  weighing  from  fifteen  to 
twenty-five  pounds,  are  used,  and  give  good  results,  since,  in  light  work,  small  rivets  are,  or  ought  to  be.  used. 
The  use  of  |-inch  steel  rivets  in  f-inch  and  even  yfinch  metal  is,  however,  not  uncommon.  It  is  very 
difficult  to  replace  loose  rivets  in  such  cases,  since  the  material  is  apt  to  split  before  the  rivet-heads  will  shear. 
Then  in  backing  the  rivet  out,  unless  the  holes  match  well,  the  material  will  be  badly  bent. 


MANUFACTURE  AND  INSPECTION  OF  IRON  AND  STEEL  STRUCTURES.  507 


Material  lighter  than  f  inch  does  not  work  up  well  in  the  shop. 

In  marking  rivets  to  be  cut  out,  the  inspector  should  use  a  centre  punch  or  the  stamping  end  of  his 
hammer  with  which  to  cut  the  head  of  the  rivet,  which  should  then  be  painted  with  white  lead.  Some  mark 
should  also  be  made  on  the  material  near  the  rivet  so  that  he  may  be  able  to  find  and  test  the  new  rivets. 

In  facing  and  boring,  care  should  be  taken  that  the  ends  of  girders  are  planed  to  the  proper  length  and 
bevel,  and  that  the  pin-holes  are  of  the  proper  size  and  distance  apart  centre  to  centre.  These  are  all  subject 
to  careful  measurements,  which  should  be  taken  with  a  steel  tape  after  being  compared  with  the  company's 
standard,  the  correction  for  every  ten  feet  being  determined. 

The  inspector  should  supervise  the  laying  out  of  the  sections  that  are  to  be  fitted  up  in  the  shop,  and 
see  that  everything  goes  together  so  that  no  unnecessary  work  has  to  be  done  in  the  field. 

When  he  receives  invoices  of  shipments  he  should  be  able  to  check  of!  all  of  these  pieces  from  those 
marked  "  Accepted  "  in  his  note-book. 

Such  a  rigid  record  of  the  work  as  outlined  above  requires  some  office  work  which  has  to  be  done  at 
odd  times  or  in  the  evening.  But  it  pays  for  itself  many  times  over  in  the  sense  of  absolute  security  which 
the  inspector  is  thereby  enabled  to  enjoy. 


5o8 


APPENDIX, 


APPENDIX  C. 
AMERICAN  METHODS  OF  BRIDGE  ERECTION.* 

Bridge  erection,  in  a  broad  sense,  includes  the  assembling  in  place,  connection  and  adjustment  of  almost 
all  framed  and  trussed  structures,  chiefly  bridges  and  roofs,  either  permanent  or  temporary,  primary  or 
auxiliary ;  but  in  this  part  of  this  country,  and  as  the  most  developed  art,  it  refers  chiefly  to  large  structures 
composed  of  iron  or  steel  members,  with  which  we  may  properly  deal  from  the  time  they  leave  the  manufac- 
tory until  the  final  inspection  and  acceptance  by  the  purchaser's  engineer. 

The  subject  has  three  principal  divisions:  first.  Primary  Structures,  usually  permanent;  second. 
Auxiliary  Structures,  usually  temporary  ;  third.  Working  Plant. 

Bridges  may  be  assumed  to  include  all  structures  designed  to  transmit  strains  of  flexure  to  relatively 
solid  seats,  and  thus  embrace  roofs,  girders,  highway  and  railroad  bridges,  viaducts,  aqueducts,  towers^ 
columns,  and  wind  and  crane  bracings  that  form  most  of  the  first  division.  Primary  structures  may  be 
either  simple  or  compound;  a  simple  structure  practically  consisting  of  a  single  piece,  as  a  column  or  plate 
girder;  simple  structures  may  be  either  directly  placed  or  temporarily  supported. 

Compound  structures  may  be  very  elaborate,  like  the  complicated  trusses  of  a  long-span  railroad  bridge, 
and  are  essentially  structures  formed  by  assembling  several  members  or  parts  delivered  separately  at  the 
site.  Compound  structures  may  be,  during  erection,  naturally  self-supporting,  artificially  self-supporting,  or 
non-self-supporting.    Non-self-supporting  structures  may  be  erected  on  the  ground  or  on  falsework. 

Auxiliary  structures  are  chiefly  designed  solely  for  the  erection  of  permanent  constructions,  of  which 
they  may  serve  the  whole  or  portions;  they  may  be  fixed  or  movable.  Fixed  structures  include  trestling, 
towers,  piles,  framed  trusses,  and  suspended  platforms.  Movable  structures  include  shear-legs,  gin-poles, 
derricks,  rolling  towers  and  platforms,  and  boats. 

The  present  American  practice  is  notably  superior  to  the  foreign  in  the  completion  of  menibers  by 
power  tools  in  the  manufactories,  their  design  with  special  reference  to  rapidity  and  accuracy  of  field  assem- 
bling and  completion  of  joints,  and  for  the  liberal  use  of  special  engines  and  steam  and  hydraulic  power  in 
the  field  that  was  promoted  by  the  magnitude  and  economy  of  American  work. 

In  the  admirable  monograph  on  American  Bridges  presented  by  Ti)eodore  Cooper  to  the  Am.  Soc. 
C.  E.,  the  development  of  long  spans  and  consequently  of  heavy  members  and  difficult  erection  problems  is 
traced,  and  it  is  shown  that,  except  some  moderately  long  timber  spans,  no  great  and  heavy  trusses  existed 
until  recent  years,  so  that  their  erection  is  the  art  of  this  quarter  century  and  its  most  able  masters  are  of  the 
present  generation,  who  have  created  methods  and  appliances  at  least  as  fast  as  the  designing  engineers  and 
manufacturers  have  furnished  them  with  structures  of  increasing  proportions  to  handle. 

The  first  wooden  bridges  were  doubtless  built  on  continuous  timber  scaffolds,  each  moderate-sized  piece 
being  framed  on  the  spot  and  readily  placed  in  position  by  hand  tackle,  levers,  and  skids;  and  as  the  light 
highway  iron  work  of  twenty  or  thirty  years  ago  was  introduced,  old  methods  were  modified  to  suit.  When 
railroad  bridges  became  important,  the  erection  of  almost  every  structure  of  magnitude  was  a  problem 
requiring  special  solution,  and  new  methods  and  tools  have  been  constantly  devised,  modified,  and  perfected 
until  the  mechanical  and  constructive  skill,  ability,  and  facilities  now  acquired  are  probably  unparalleled  in 
the  world's  development  of  physical  undertakings  of  magnitude. 

The  erection  of  simple  structures  considers  chiefly  girders,  roof  trusses,  and  columns.  Girders  vary 
from  the  dimensions  of  rolled  I  beams  to  those  of  solid  plate  girders  more  than  120  feet  long  and  weighing 
over  100,000  lbs.  each,  or  of  lattice  girders  of  1 50  feet  or  more  in  length,  the  girders  up  to  120  feet  long  having 
been  shipped  from  the  shops  in  single  rigid  pieces.  Such  long  and  heavy  pieces  must  be  loaded  skilfully  to 
ride  the  railroad  curves,  and  each  requires  from  three  to  five  flat-cars  for  its  transportation.  The  girder  is 
supported  at  each  end  on  a  transverse  beam  that  has  an  iron  bar  or  old  rail  on  top  on  which  the  girder  rests, 
and  can  easily  slip  to  conform  to  the  chords  of  the  curves.  This  transverse  beam  is  supported  by  the 
centres  of  two  or  more  longitudinal  ones  whose  ends  rest  on  transverse  beams  placed  on  the  car  floor  and 


•'•  Adapted  from  an  illustrated  lecture  delivered  by  Mr.  Frank  W.  Skinner,  M.  Am.  Soc.  C.  E.  and  of  the  editorial 
staff  of  The  Engineering  Record,  before  the  College  of  Civil  Engineering  of  Cornell  University,  April,  1893. 


AMERICAN  METHODS  OF  BRIDGE  ERECTION. 


thus  distributing  the  load  on  two  lines,  one  at  each  end  of  the  car,  or  else  rest  on  another  set  of  longitudinals 
that  are  set  on  four  transverse  beams,  one  of  which  would  thus  be  directly  over  each  axle  and  sustain  one 
eighth  of  the  total  load,  half  of  wiiich  is  carried  by  each  end  car,  the  intermediate  ones  acting  only  as 
spacers.  When  there  is  sufficient  head  room  girders  may  be  loaded  edgewise,  but  otherwise  and  more  often 
they  are  loaded  flatwise.  Whenever  practicable,  they  are  not  unloaded  until  brought  across  the  openings 
they  are  intended  to  span  and  parallel  to  their  final  positions  from  which  they  do  not  vary  longitudinally  and 
not  more  than  is  necessary  transversely.  They  are  then  usually  raised  a  little  by  hydraulic  jacks  and  sup- 
ported by  timber  blocking  till  the  cars  are  run  out  from  beneath  them  and  then  jacked  down  and  skidded  tc 
their  seats,  or,  less  frequently,  are  lifted  from  gallows  frames  and  turned  if  necessary  and  lowered  by  tackle. 
These  gallows  frames,  one  at  each  end,  ordinarily  consist  of  single  bents  of,  say,  12x12  posts  and  single  or 
reinforced  caps  that  just  span  one  or  two  tracks  and  are  guyed  both  ways.  When  no  old  or  temporary  track 
exists  across  the  opening,  the  girders  have  been  unloaded  at  one  end  of  it  and  placed  in  the  required  posi- 
tion by  protrusion,  i.e.,  pushed  out  cantilever-wise  over  a  stationary  roller  on  the  abutment  until  the  forward 
end  reached  its  seat  on  the  opposite  side.  This  method  requires  eitlier  a  pilot  extension,  a  rear  counter- 
weight, overhead  guys  or  intermediate  supporting  rollers;  after  it  is  half  way  across  a  pilot  extension  would 
generally  be  used  and  would  be  a  long  beam  lashed  firmly  to  the  girder  so  as  to  engage  a  roller  on  the 
further  side  before  the  centre  of  gravity  of  the  girder  passed  the  first  abutment. 

A  remarkable  example  of  this  method  of  erection  is  that  of  the  Souleuvre  Viaduct  in  France  ;  its  spans, 
of  riveted  lattice-girder  type,  were  completely  assembled  about  i6oo  feet  from  one  abutment,  connected 
together  as  continuous  spans,  and  rolled  out  on  fixed  rollers;  the  spans  weighed  nearly  100,000  lbs.  each  and 
had  in  front  a  trussed  pilot  66  feet  long  that  weighed  40.000  lbs.  The  bridge  was  protended  by  the  revolu- 
tion of  the  fixed  rollers  at  each  pier.  These  rollers  were  turned  by  ratchets  operated  by  long  levers,  one  on 
each  side  of  the  span,  connected  by  a  cross-bar  over  the  top  of  the  bridge.  Men  walking  back  and  forth  on 
the  top  of  the  bridge  pushed  this  cross-bar  before  them  and  thus  turned  the  rollers,  but  considerable 
difficulty  was  experienced  in  securing  uniformity  of  action  between  different  gangs.  This  example  is  notable 
in  that  it  should  have  been  successful,  and  for  its  striking  difference  from  American  practice. 

Roof  trusses  up  to  about  100  feet  span  are  generally  lifted  and  set  in  place  as  one  complete  finished 
truss,  whether  with  rigid  or  flexible  joints.  If  with  riveted  joints,  they  have  been  shipped  from  the  shops  in 
one,  two,  three,  or  four  sections  each,  that  are  riveted  together  on  the  ground  at  the  site;  or,  it  pin-con- 
nected, they  are  assembled  there,  and  in  either  case  raised  and  set  by  a  gin-pole  or  derrick  that  moves  back- 
ward with  each  successive  truss. 

These  trusses  often  depend  largely  upon  the  roof  sheathing  boards  for  lateral  bracing,  without  which  they 
have  little  transverse  stiffness.  Tiiey  are  also  likely  to  be  set  on  slender  isolated  columns,  and  require 
special  care  in  guying  until  permanently  braced  after  being  released  from  the  derricks.  When  supported  on 
columns  the  trusses  may  be  assembled  between  them  exactly  parallel,  and  each  with  its  lower  chord  nearly 
in  the  vertical  plane  of  its  final  position,  but  if  supported  on  masonry  walls  they  must  be  assembled  with 
their  lower  chords  sufficiently  oblique  to  clear  and  be  adjusted  after  rising  above  the  tops  of  the  walls.  They 
may  also  be  assembled  on  the  same  platform  or  blocking  at  one  end  of  the  building  and  raised  there  without 
moving  the  derrick,  and  skidded  along  on  top  of  the  walls  to  their  respective  positions ;  but  this  method  will 
usually  be  more  difficult,  tedious,  and  hazardous  than  moving  the  derrick  to  raise  them  in  position.  This 
method  was  employed  at  one  of  the  mills  in  the  famous  Homstead  Plant,  and  either  one  end  of  a  certain 
truss  was  advanced  beyond  the  other,  or  else  the  flexibility  was  so  great  that  it  was  pulled  out  of  a  plane  and 
the  lower  chord  became  curved  horizontally  so  that  it  fell  off  and  tumbled  to  the  ground. 

Two  gin-poles  are  often  used  togetlier  to  raise  a  roof  truss,  each  grippiifg  it  about  one  fourth  or  one  fifth 
of  its  length  from  the  centre,  and,  of  course,  always  above  its  centre  of  gravity.  The  writer  once  used  this 
method  of  erecting  a  rolling-mill  roof  of  over  140  feet  spans  whose  very  light  trusses  had  slender  gaspipe 
struts  and  deck-beam  top  chords,  and  were  about  limber  enough  to  be  considered  funicular  machines,  but 
were  handled  without  difficulty  by  the  renforcement  of  planks  judiciously  lashed  on. 

A  gin-pole  is  simply  a  timber  mast  with  four  guys  and  a  sheave  at  the  top  over  which  the  hoist  line  leads 
to  a  crab  bolted  three  or  four  feet  from  the  bottom.  In  use  it  should  always  be  inclined  a  little  from  the 
vertical  so  that  it  overhangs  its  burden  and  gives  a  positive  strain  on  the  back  guys  and  on  them  only,  the 
front  guys  coming  into  service  when  the  pole  is  moved.  By  taking  up  or  slacking  the  guys  the  truss  may  be 
very  quickly  swung  backwards,  forwards,  or  transversely  and  adjusted  to  position,  and  for  heavy  work  a 
tackle  is  advantageously  used  to  operate  at  least  the  back  guys. 

The  foot  of  a  gin-pole  is  generally  supported  by  and  shifted  upon  a  plank  or  timber  along  which  it  is 
pinched  with  bars  or  pushed  upon  rollers.  Gin-poles  are  ordinarily  from  40  to  60  feet  long  and  up  to  16 
inches  square  at  the  butt.  In  erecting  a  lofty  dome  recently  Horace  E.  Horton,  of  the  Chicago  Bridge 
Works,  used  a  trussed  gin-pole  120  feet  long.    Gin-poles  are  often  rigged  with  J-inch  wire  guys  and  i^-inch 


APPENDIX. 


manilla  line  that  would,  according  to  the  load,  be  operated  directly  by  the  crab  or  be  rove  over  a  fixed  and 
loose  single  block  or  a  two-three  pair  of  blocks. 

An  A  derrick  is  two  inclined  masts  braced  together  and  united  at  the  top;  needs  but  one  guy  and  is 
very  often  preferable  to  a  gin-pole. 

A  gin-pole  must  always  be  carefully  handled  and  may  be  easily  raised  or  lowered  by  a  boom  or  an  A 
derrick  that  is  likely  to  be  found  at  any  large  building,  provided  the  height  of  the  pole  is  not  more  than 
twice  that  of  the  derrick,  which  can  then  pick  it  up  just  above  its  centre  of  gravity,  and  swing  it  into  its 
vertical  position.  If  the  gin-pole  is  only  a  little  too  long  for  this  it  can  still  be  handled,  as  by  counterweighing 
the  butt.  A  short  pole  can  be  raised  by  fastening  the  foot  and  blocking  the  other  up  beyond  the  centre  of 
gravity  until  the  angle  is  great  enough  to  enable  it  to  be  revolved  by  a  rope  leading  to  the  ground,  but  much 
the  easiest  and  best  way  is  to  provide  a  secure  resistance  for  the  foot,  making  a  virtual  hinge  there,  and 
hoist  it  from  the  ground  by  a  line  led  over  the  top  of  a  shears,  which  need  only  be  any  convenient  pair  of 
timbers  lashed  together  at  the  top.  When  the  pole  rises  high  enough  to  carry  the  rope  ofif  from  the  shears 
they  will  be  no  longer  needed,  and  if  the  hoisting  crab  has  been  properly  set  it  will  continuously  pull  the 
pole  up  to  its  vertical  position.  The  foot  of  the  pole  must  be  carefully  watched,  and  reliable  men  stationed 
at  the  guys  which  always,  whether  moving  or  raising  the  pole,  should  be  kept  free  from  slack. 

When  the  span  is  very  great,  or  when  the  ground  underneath  must  be  preserved  free  from  obstruction, 
the  roof  trusses  are  either  assembled  and  erected  from  a  strident  traveller,  or  a  tower,  or  upon  a  movable 
platform  whose  surface  conforms  to  its  lower  chord  or  intrados,  is  as  long  as  the  span,  and  usually  is  a  little 
wider  than  the  distance  from  out  to  out  of  the  two  most  distant  adjacent  trusses,  so  that  each  pair  may  be 
simultaneously  erected  in  position,  braced  together,  and  left  in  stability  while  it  moves  forward  two  panels' 
lengths  to  the  next  pair. 

Lofty  viaducts  have  been  erected  with  the  utmost  simplicity  by  booms  carried  on  the  structure  itself  as 
its  towers  were  built  up  section  by  section  from  the  ground. 

The  famous  Kinzua  viaduct,  over  300  feet  high,  was  erected  in  this  manner  several  years  ago,  but  the 
method  is  evidently  best  adapted  to  locations  where  the  iron  work  is  most  readily  delivered  in  the  bottom 
of  the  chasm,  which  is  not  usually  the  case,  and  probably  would  not  now  be  used  except  for  very  short  struc- 
tures or  where  the  spans  between  towers  were  extremely  long,  the  material  being  chiefly  handled  from  over- 
head in  the  present  practice. 

By  far  the  most  usual  and  generally  economical  way  of  erecting  viaducts,  including  metropolitan 
elevated-railroad  structures,  is  by  means  of  an  overhanging  derrick  that  moves  on  top  of  the  completed 
portion  and  reaches  far  enough  beyond  it  to  set  all  members  in  the  next  one  or  two  panels  in  advance  of  its 
own  support,  setting  and  maintaining  each  piece  until  it  is  braced  and  self-supporting;  the  connections 
usually  being  quickly  and  temporarily  bolted  up  to  enable  the  derrick  to  move  forward  onto  the  new  panel 
and  commence  erecting  the  next  one  before  the  main  joints  are  drifted  and  riveted  and  secondary  connec- 
tions completed.  These  derricks  are  called  "  Erecting  Travellers,"  and  comprise  essentially  a  base  moving 
on  the  finished  work,  and  carrying  the  hoisting  engine,  coal,  etc.,  that  partly  counterbalance  the  overhang 
and  its  burden.  A  reach  of  60  feet  usually  suffices  for  elevated  railroads  whose  travellers  may  consist  simply 
of  long,  single  beams,  mounted  on  central  wheels  and  set  with  the  front  end  slightly  elevated  and  the  rear  or 
trailing  end  lashed  or  otherwise  secured  to  the  longitudinal  girders  as  was  done  on  some  of  the  earlier  Brr-ok- 
lyn  work.  Generally,  however,  a  braced  platform,  of  the  full  width  of  the  structure,  carries  a  vertical  head 
frame  in  which  are  set  masts  ot  two  or  three  boom  derricks,  whose  booms  are  usually  trussed  and  swing 
nearly  around  a  semi-circle  so  as  to  be  able  to  pick  up  the  iron  from  the  side  of  the  street  and  swing  it  into 
position.  Often  these  booms  are  arranged  so  that  the  two  side  ones  can  set  and  hold  the  columns  of  the 
next  panel  while  the  longer  centre  boom  puts  the  transverse  girder  in  position  upon  them  and  after  it  is 
connected  to  them  maintains  the  whole  bent  until  the  released  side  booms  set  the  longitudinal  girders  and 
make  the  whole  panel  stable. 

Railroad  viaducts  are  generally  considered  to  be  most  economically  proportioned  when  the  towers  are 
30  feet  wide  and  60  feet  apart,  and  many  have  been  built  approximately  to  these  dimensions,  thus  requiring 
an  overhang  to  support  a  transverse  beam  or  pair  of  column  sections  at  a  distance  of  90  feet,  a  short  longi- 
tudinal girder  at  75  feet,  and  a  long  one  at  30  feet.  These  travellers  always  consist  of  two  parallel  trusses, 
jsually  combination,  always  firtnly  braced  together  horizontally  and  anchored  to  the  structure  when  in 
service.  Sometimes  the  overhangs  are  single  heavy  beams  with  iron  guys  from  the  end  and  intermediate 
points  to  the  top  of  a  mast  placed  over  the  end  of  the  supported  part,  and  guyed  back  to  the  rear  of  the  plat- 
form. But  they  are  more  often  square  Howe  or  Pratt  trusses  not  unlike  bridge  spans,  overhanging  about 
half  their  length,  and  having  cross-beams  and  eye-bolts  in  the  lower  chords  from  which  to  suspend  the 
tackles  required  to  lift  and  support  the  pieces  in  the  different  positions  of  the  towers  and  spans. 

Viaduct  travellers  are  designed  according  to  special  conditions  so  as  to  receive  the  members  for  erection 


Plate  XLIV. 


AMERICAN  METHODS  OF  BRIDGE  ERECTION.  S»l 

directly  underneath  the  overhang,  or  from  cars  that  run  on  its  own  track  level  and  come  from  behind  up  to 
or  underneath  the  main  platform  and  deliver  to  trolleys  that  carry  the  hoisting  tackle  out  on  the  overhang, 
or  to  booms  that  swing  it  around,  or  to  falls  that  slack  it  off  to  position.  In  building  the  St.  Paul  High 
Bridge  across  the  Mississippi  River,  Horace  E.  Horton  erected  the  long  and  lofty  viaduct  by  a  huge  trav- 
elling tower  that  was  150  feet  high  by  68  feet  square,  and  straddled  the  125  feet  high  trestle  bents,  with  a 
clear  spring  of  more  than  135  feet  high.  It  was  built  chiefly  of  5"  x  10"  and  smaller  sizes  of  timber 
with  iron  main  tension  diagonals;  ran  on  eight  double-flanged  wheels,  and  was  probably  the  largest  and 
tallest  traveller  ever  constructed. 

As  crane  bracing  and  horizontal  trusses  must  be  permanently  supported  by  columns,  walls  or  vertical 
trusses,  the  supports  greatly  facilitate  their  convenient  erection,  which  is  almost  invariably  accomplished  by 
simple  tackle  directly  supported  and  by  stationary  crabs  or  engines. 

The  only  primary  structures  remaining  to  be  considered  are  Long  Span  Bridges,  i.e.,  say  above  150  feet 
long.  They  may  liave  either  pin  or  riveted  connections,  or  be  suspension  bridges  or  arches.  Most  of  them 
in  this  country  are  the  former,  although  there  is  a  growing  tendency  to  design  arches  for  locations  where  the 
geological  formation  affords  good  seats  that  in  receiving  the  thrust  save  the  metal  required  for  a  tension 
chord. 

Cantilevers  and  suspension  bridges  are  the  only  types  that  are  self-supporting  during  erection.  The 
former  may  include  draw  bridges  when  they  are  erected  symmetrically  with  the  panels  simultaneously  added 
each  side  of  the  centre  so  as  to  balance  each  other  upon  the  pivot  pier,  but  they  are  generally  erected  upon 
the  fender  piling  in  the  axis  of  the  river  each  side  of  the  pier. 

In  cantilevers  the  anchor  arm  is  first  built  on  falsework  and  counterweighted  so  as  to  enable  the 
channel  arm  to  be  built  as  an  overhang,  its  members  being  self-sustaining  as  soon  as  each  panel  is  con- 
nected. 

American  cantilevers  are  almost  invariably  connected  by  a  suspended  centre  span  of  from  i  to  |  of  the 
total  opening,  and  this  is  usually  erected  as  an  extension  of  the  cantilever  arms  from  each  side,  special  tem- 
porary or  permanent  stock  being  provided  in  the  truss  members  if  necessary  to  meet  the  erection  strains, 
whicli  are  usually  allowed  a  high  unit  value. 

Wire  Suspension  Bridges  are  commonly  erected  from  their  own  cables,  which,  when  twisted  rope  is 
used,  are  drawn  across  the  river  in  strands  and  then  lifted  to  the  tops  of  the  towers. 

The  large  cables  are  merely  bundles  of  parallel  straight  wires  that  are  carried  across  singly  by  special 
machines,  looped  over  the  end  pins,  and  spliced  at  the  end  of  each  coil  so  as  to  form  one  practically  contin- 
uous filament,  the  different  individual  catenaries  of  which  must  be  carefully  adjusted  and  secured  to  uniform 
tension  and  then  compacted  and  encased.  After  the  cables  are  completed  the  members  of  the  floor  and 
stiffening  trusses  and  working  derricks  and  platforms  are  supported  readily  from  them. 

Arches  are  generally  assembled  on  falsework,  but  may  be  sustained  without  it  by  commencing  at  the 
skew-backs  and  supporting  each  section  by  overhead  guys,  as  was  notably  done  in  Eads's  St.  Louis  Bridge. 
Such  a  method,  or  that  of  temporary  reinforcements  to  enable  an  ordinary  truss  to  be  erected  cantilever-wise, 
may  be  termed  artificial  self-support. 

Tubular  Bridges  are  happily  obsolete  in  this  country,  the  only  important  one  in  America  being  the 
Victoria  Bridge  across  the  St.  Lawrence  at  Montreal.  It  has  many  long  spans,  over  deep  and  rapid  water, 
the  bottom  is  too  rocky  for  pile  driving,  and  the  writer  has  been  informed  that  the  superstructure  was  built 
in  situ  in  the  winter,  upon  falsework  erected  on  the  ice  which  forms  and  packs  and  freezes  there  to  remark- 
able thicknesses  so  that  higher  up  on  the  river  it  is  not  uncommon  to  run  ordinary  locomotives  across  on  it. 
Stevenson's  other  great  tubular  bridges  at  Conway  and  the  Straits  of  Menai,  England,  were  built  at  the 
water's  edge,  floated  complete  to  the  unfinished  pier,  and  raised  many  feet  to  final  position  by  hydraulic 
jacks,  their  masonry  seats  being  continually  built  up  close  under  them  as  they  rose. 

With  the  above  exceptions,  long-span  arches  and  trusses  are  supported  on  falsework  substructures  until 
self-sustaining. 

Fixed  Falsework. 

This  may  consist  of  a  simple  temporary  suspension  bridge  with  a  more  or  less  stifTened  floor  for  com- 
paratively light  work  or  where  the  height  or  cost  of  falsework  would  be  prohibitory,  or  where  it  could  not 
be  safely  maintained.  Excellent  examples  are  afforded  by  the  South  American  work  of  L.  L.  Buck,  who 
erected  some  of  the  first  railroad  bridges  in  the  Andes  mountains,  with  the  simplest  equipment  and  unskilled 
labor;  the  structures  spanned  deep  chasms  traversed  by  mountain  torrents  that  were  liable  to  sudden  floods 
that  would  destroy  trestle  falsework,  and  the  trusses  could  not  be  erected  cantilever-wise.  A  suspension 
bridge  was  therefore  made  across  the  river,  and  on  its  floor  the  trusses  were  erected  by  a  light  special  derrick 


APPENDIX. 


that  commenced  at  the  centre  and  erected  to  one  end  by  balanced  overhangs  that  were  lowered  to  allow  it  to 
return  inside  the  erected  structure  to  the  centre  and  thence  erect  the  remainder  of  the  truss. 

In  an  adjacent  viaduct  crossing,  a  cable  was  stretched  from  side  to  side,  and  upon  it  a  trolley  hoist 
travelled  by  gravity,  receiving  the  iron  members  on  top  of  the  bank  and  lowering  them  as  it  carried  them 
out  to  position  so  that  the  resultant  motion  carried  them  in  a  nearly  straight  line  to  their  required  locations 
ill  the  towers. 

Reverse  conditions  existed  at  the  Elkhorn  viaduct  erected  recently  by  CofTrode  &  Evans,  Philadelphia. 
The  timbers  for  a  600-foot  permanent  viaduct  140  feet  high  were  delivered,  framed  and  connected  up  in  sec- 
tions of  trestle  bents,  at  the  centre  of  the  bottom  of  the  chasm  where  a  stationary  hoisting  engine  raised 
them,  traversed  them  along  the  line  of  the  structure,  and  set  them  in  position  by  means  of  a  trolley  travelling 
on  a  suspension  cable  and  operated  by  hoist  and  traverse  lines  leading  from  it  to  snatch-blocks  at  the  foot 
of  the  opposite  towers. 

Trussed-span  falsework  has  been  used  chiefly  where  there  was  great  danger  from  floods,  drift  or  scour, 
or  where  it  was  imperative  to  oiler  the  least  obstruction  to  navigation. 

Howe  trusses  or  combination  trusses  have  generally  been  used;  and  can  be  quickly  erected  on  trestles 
to  seat  on  the  bridge  piers  and  become  self-sustaining,  and  permit  the  removal  of  the  trestles  and  allow 
plenty  of  time  for  the  removal  of  old  and  assembling  of  new  structures. 

The  long  spans  of  one  of  the  first  Missouri  River  bridges  were  erected  on  one  temporary  Howe  truss 
deck  span  that  was  assembled  on  trestle  work  of  the  required  height,  then  transferred  to  two  wooden  towers 
built  on  the  decks  of  pontoons  which  were  floated  with  their  burden  to  position  between  the  piers  ;  water 
was  then  admitted  to  the  pontoons  and  they  sunk  sufficiently  to  deposit  the  span  on  the  piers  and  clear  it 
and  be  removed,  leaving  a  platform  safe  from  floods,  for  the  assembling  and  connecting  at  leisure  of  the 
permanent  span.  After  it  was  swung  the  pontoons  and  towers  were  brought  back  underneath,  pumped  out 
till  their  buoyancy  lifted  the  Howe  span,  which  was  then  transferred  to  the  next  opening  and  seated  as 
before,  for  the  erection  of  another  span,  and  so  on. 

This  erection  was  notable  for  the  length  of  the  wooden  span,  the  height  of  the  towers,  the  swiftness  of 
the  current,  and  tlie  success  and  economy  of  the  operation. 

Several  years  after  this  erection  was  that  of  the  bridge  of  the  Canada  Atlantic  Railroad  across  the  St. 
Lawrence  River  at  the  head  of  the  Coteau  rapids,  with  a  355-ft.,  swing  span  and  one  139-ft.,  two  175-ft., 
four  223-ft.,  and  ten  217-ft.  fixed  spans.  The  water  was  from  20  to  30  feet  deep,  with  a  current  of  5  to  7 
miles  an  hour,  making  it  impossible  to  erect  falsework  on  the  rocky  bottom,  and  very  difficult  to  build  the 
masonry  piers,  which  were  constructed  in  bottomless  wooden  caissons  that  were  floated  into  position,  loaded 
and  sunk  with  inside  canvas  bottom  fiaps  on  which  the  concrete  was  laid. 

Three  miles  above  the  site,  in  a  sheltered  bay,  falsework  was  built  parallel  to  the  shore,  and  at  its  ends 
transverse  tracks  were  built  out  to  deeper  water.  Between  these  tracks  and  parallel  to  the  falsework  were 
moored  a  pair  of  timber  pontoons  each  90  x  26  x  6  feet  deep,  and  braced  together  70  feet  apart.  On  the 
deck  of  each  pontoon  a  trestle  a  little  higher  than  the  bridge  piers  was  built.  The  permanent  spans  were 
assembled  and  connected  complete  on  the  falsework,  skidded  down  the  transverse  tracks  above  the  towers, 
which  rose  and  lifted  them  free  when  the  water  was  pumped  out  of  the  pontoons,  and  supporting  them  at 
two  panels' distance  from  each  end  were  slacked  ofT  down  the  river  and  sunk  between  piers  enough  to  deposit 
the  span  on  its  seats,  after  which  they  returned  for  another  span,  and  so  on. 

In  the  erection  of  the  Harvard  Bridge  at  Boston,  the  long  plate-girder  spans  were  built  on  shore,  lifted 
by  a  traveller  that  overreached  their  ends  and  at  high  tide  carried  out  by  it,  deposited  on  a  special  pontoon 
that  was  towed  to  position,  and  with  the  falling  tide  deposited  the  girders  on  their  pier  seats. 

The  Brunot's  Island  Bridge  near  Pittsburgh,  and  the  Hawkesbury  Bridge  in  Australia  erected  by 
C.  L.  Strobel  and  Charles  McDonald,  are  notable  illustrations  of  moving  very  large  spans  at  a  great  height 
upon  pontoons. 

Trestles. — These  are  essentially  sets  of  columns  in  vertical  planes  transverse  to  the  axis  of  the 
structure;  they  are  in  rows  or  bents  of  four  or  more  with  transverse  and  longitudinal  bracing,  the  latter 
either  continuous  or  connecting  alternate  pairs  so  as  to  form  towers.  When  the  height  is  considerable  the 
trestles  are  built  in  stories  so  as  to  give  convenient  sections.  The  simplest  frami  d  trestiing  has  bents  each 
composed  of  two  plumb  posts  and  two  batter  posts,  a  cap,  a  sill,  and  two  diagonal  braces,  and  the  bents  are 
braced  longitudinally  by  a  horizontal  ledger  piece  at  the  top  and  a  diagonal  from  the  top  of  one  to  the  foot 
of  the  next,  on  each  side.  Usually  the  caps  are  connected  by  two  or  more  lines  of  stringers  which  should 
rest  directly  above  the  adjacent  tops  of  the  plumb  and  batter  posts,  the  latter  having  a  run  of  i  in  12  to  i  in 
4  horizontal  to  vertical  according  to  circumstances.  The  dimensions  of  the  timber  should  be  proportioned 
to  the  load  carefully  in  high  or  exposed  structures  or  for  very  heavy  burdens,  but  must  never  be  made  very 
light,  for  the  connections  never  develop  all  the  efficiency  of  the  sections,  and  weight  and  stiffness  are  gener- 


AMERICAN  METHODS  OF  BRIDGE  ERECTION: 


5*3 


ally  more  important  than  direct  theoretical  strength,  so  that  careful  judgment  and  experience  are  much 
more  reliable  and  necessary  than  elaborate  calculation  of  strains. 

For  ordinary  simple  trestles  lo  x  lo  and  12x12  posts  and  caps,  3  x  10  or  3  x  12  braces,  and  8  x  16 
stringers  are  much  used,  and  for  lofty  and  elaborate  structures  the  variation  is  more  in  the  number  and 
arrangement  of  pieces  than  in  their  sections,  though  some  6  x  6  and  8x8  are  used  for  secondary  braces, 
and  larger  ones  for  important  members  if  obtainable.  But  the  few  sizes  mentioned  with  2-inch  platform 
plank  and  plenty  of  f  and  f  bolts,  large  washers,  wood  screws,  and  steel  spikes  comprise  most  of  the  bill  of 
materials,  except  occasionally  screw-ended  iron  tension  rods  with  cast  angle  blocks.  Mortise  and  tenon 
foints  are  seldom  used  ;  sometimes  a  batter  post  is  toed  in,  but  most  joints  are  butted  and  covered  with  a 
splice  or  "  batten  "  piece  each  side  ;  usually  a  2-  or  3-inch  plank  as  viide  as  the  timber,  and  from  4  to  6  feet 
long,  secured  with  four  or  more  bolts. 

Sometimes  double  caps  are  used,  one  piece  on  each  side  of  the  posts,  but  usually  single  pieces  of  the 
same  thickness  as  the  posts  are  put  in  their  vertical  plane,  and  between  the  tops  of  the  lower  and  the 
bottoms  of  the  upper  sections,  and  all  are  bound  together  by  batten  pieces  across  the  caps. 

Trestle  bents  may  be  from  5  to  20  feet  apart  and  of  any  height  from  4  to  150  feet.  Smaller  heights  are 
blocked  up  solid,  and  greater  ones  are  usually  avoided  or  otherwise  provided  for.  Stories  do  not  ordinarily 
exceed  20  feet  in  height.  Ledger  pieces  and  diagonals  are  sometimes  spiked  on,  especially  to  promote 
rapidity  of  erection,  but  the  latter  at  least  should  generaly  be  bolted  eventually. 

When  framed  trestles  are  to  be  set  with  sills  on  a  river  bottom,  careful  soundings  must  first  be  taken 
preferably  with  a  pole,  and  a  profile  made  to  determine  the  heights  of  the  posts.  The  bents  are  laid  out  with 
uniform  tops  and  bottom  widths  varying  to  suit  the  different  heights  and  bottoms.  Tiiey  are  bolted  together 
on  shore,  floated  to  position,  swung  up  and  set  vertically  from  a  derrick  or  pile  driver  boom  or  from  the  cap 
of  the  last  bent  if  it  is  high  enough,  and  stayed  by  ledgers,  braces,  or  stringers. 

If  the  water  is  deep,  the  lower  end  of  the  longitudinal  diagonal  brace  is  bolted  to  the  foot  of  the  batter 
post  before  the  bent  is  set,  so  as  to  easily  bring  it  eventually  at  the  desired  position  under  water. 

Sometimes  the  bottom  is  explored  ahead,  or  the  bents  raised  froni  a  "  balance  beam,"  which  is  simply  a 
long  heavy  timber  projecting  half  its  length  beyond  the  last  trestle  set,  upon  whose  cap  its  middle  is  sup- 
ported, while  its  rear  end  is  under  the  cap  of  the  preceding  bent. 

Ordinarily  the  "mud  sill"  is  a  sufficient  footing  when  framed  trestles  are  admissible,  but  sometimes 
special  provisions  are  requisite  on  account  of  severe  burdens  or  very  soft  bottom.  In  trestling  over  a  very 
soft  mud  John  Devin  secured  very  cheap  and  efficient  footings  by  spiking  old  railroad  ties  transversely  to 
the  bottom  of  the  sills  of  his  trestles,  and  thus  constructing  what  was  substantially  a  grillage  under  each 
bent. 

Pile  Trestles. — Wherever  piles  can  penetrate  they  form  the  best  foundations  for  falsework.  They 
are  driven  to  a  moderate  refusal  and  loaded  much  more  heavily  than  in  permanent  structures.  They  are 
used  in  single  lengths  up  to  about  60  feet,  beyond  which  they  must  usually  be  spliced,  the  joint  being  gener- 
ally a  square  butt  with  comparatively  narrow  batten  splice  pieces  bolted  or  spiked  on  all  around  it.  At  the 
Poughkeepsie  Bridge  piles  were  thus  driven  120  feet  through  the  mud  and  water. 

Piles  are  driven  either  by  a  floating  driver  or  by  one  ruiming  along  on  top  of  the  comjjleted  bents. 
The  batten  posts  are  driven  by  inclining  the  ram  guides  to  correspond  to  the  required  inclination. 

If  the  top  of  the  trestle  is  within  10  or  15  feet  of  the  surface  of  the  water  the  piles  are  usually  sawed  of! 
and  capped  just  below  that  height,  but  if  the  elevation  is  much  greater  they  are  usually  capped  near  the 
water  level,  and  one  or  more  stories  of  framed  trestle  built  upon  them. 

Sometimes,  to  prevent  brooming,  piles  are  driven  with  a  temporary  iron  ring  or  ferrule  on  top;  some- 
times they  have  iron  points  or  cast  shoes  at  the  bottom. 

Great  judgment  is  required  in  driving  piles,  which  should  be  controlled  far  more  by  skilled  experience 
than  by  theoretical  considerations ;  sometimes,  when  the  penetration  is  excessive,  if  they  are  allowed  to  rest 
a  few  hours  they  will  resist  heavier  blows  of  the  ram  than  previously  drove  them,  and  be  amply  safe  for 
severe  static  loads.  At  other  times  the  pile  will  refuse  to  penetrate  and  will  bound  out  at  every  blow,  but 
will  sink  steadily  if  loaded  with  a  dead  weight  and  tapped  or  twisted. 

Some  remarkable  and  apparently  extravagant  pile  work  was  done  at  the  Gour-Noir  Railroad  viaduct 
over  the  La  Vezere  River,  France,  where  the  213-ft.  main  stone  arch  span  26  feet  wide  between  parapets  and 
5'  7"  thick  at  the  crown,  was  built  on  centring  carried  by  seven  continuous  wooden  trusses  14'  6"  deep  that 
each  rested  on  sand  boxes  on  the  tops  of  oak  piles  12  inches  or  more  in  diameter  ;  their  bottom  ends  cut 
square  and  shod  with  plate  iron.  The  river  is  subject  to  floods  and  has  a  granite  bed  into  which  holes  3  feet 
deep  were  drilled  and  the  piles  wedged  and  cemented  in  them.  Quartz  veins  were  encountered  in  which 
only  one  tenth  of  an  inch  per  hour  could  be  drilled  until  cofferdams  were  built.  Tlie  cost  of  setting  126 
piles  was  about  $6800. 


5i4 


APPENDIX. 


Timber  dimensions  for  Eastern  irestlinrr  schioin  exceed  12  and  16  or  at  the  utmost  2o  inches,  but  a 
recent  letter  from  a  bridge  man  on  the  Pacitic  coast  says  his  regular  bills  call  for  sizes  like  ii"  x  30"  x  42', 
20"  X  20"  X  48',  and  7"  x  18"  x  64',  while  he  saw  some  logs  being  sawed  into  timber  48"  x  48"  x  50'  and 
20"  X  20"  X  100'. 

Scmetimes,  to  afford  passageway  for  navigation,  a  few  bents  will  be  omitted  at  the  bottom  of  a  line  of 
falsework,  and  the  opening  will  be  bridged  by  a  high  temporary  span  to  carry  the  contmuous  top  bents. 
Such  an  opening  may  be  as  much  as  75  feet  wide  at  the  bottom  and  narrowed  at  the  top  by  inclined  posts 
so  as  to  permit  a  sort  of  queen-post  span  of  comparatively  short  pieces  or  the  use  of  stringers  borrowed  fron; 
che  permanent  structure  to  close  the  opening  at  the  top. 

In  the  early  days  of  iron  bridge  building,  it  was  frequently  customary  to  build  "  two-story  falsework," 
i.e.,  to  provide  two  complete  platforms  the  whole  length  of  the  bridge,  to  support  top  and  bottom  chords 
throughout;  but  now  it  is  almost  universal  to  support  only  the  bottom  chord  directly,  and  to  make  the  rest 
of  the  truss  self-supporting  when  assembled  upon  it,  stability  being  of  course  assured  as  soon  as  any  two 
opposite  panels  of  the  two  trusses  are  assembled  and  connected  transversely  to  each  other. 

Wooden  derricks  or  towers  usually  traverse  the  falsework  to  lift  and  place  various  members,  or  are 
fitted  to  follow  out  at  the  extremity  of  a  cantilever  arm  and  overhang  it  so  as  to  assemble  and  connect  the 
successive  panels  upon  which  it  continually  advances.  These  structures  are  called  travellers  and  properly 
belong  to  the 

Working  Plant, 

and  form  the  most  important  item  of  erection  equipment.  A  traveller  in  its  simplest  form  consists  of  one 
transverse  bent  double  guyed  or  two  bents  braced  together  longitudinally  to  form  a  tower,  each  bent  con- 
sisting of  a  post  each  side  and  a  cap  and  knee  braces  at  the  top.  Commonly,  however,  there  are  three  or  four 
bents  on  large  work,  leaving  at  each  side  a  vertical  post  and  a  batter  post  braced  together,  the  latter  flaring 
out  at  the  top  and  their  feet  united  on  a  heavy  double  longitudinal  stringer,  underneath  which  are  three  or 
more  double-flanged  wheels  that  run  on  special  tracks  laid  on  the  ends  of  the  trestle  caps ;  the  caps  of  the 
traveller  bents  are  developed  into  Howe  or  Warren  girder  trusses,  and  usually  carry  longitudinal  straining 
beams  from  which  the  hoisting  tackle  is  hung.  The  bents  are  braced  together  with  iron  or  wooden  diagonals, 
and  the  structure  carries  working  platforms  and  hoisting-engine  and  perhaps  other  machinery,  besides  some- 
times tracks  for  material  cars.  Generally  the  traveller  goes  astride  the  completed  structure  and  clears  every 
portion  of  it,  and  is  so  required  to  have  a  large  free  opening  from  end  to  end  with  no  transverse  connection 
between  the  bottoms  of  the  posts;  but  it  sometimes  is  designed  to  go  inside  the  trusses,  or  backwards  in 
advance  of  them  from  a  starting  point,  or  upon  the  top  chords,  in  which  cases  it  may  have  bottom  and 
interior  cross  and  diagonal  bracing. 

Traveller  Bents,  as  at  Poughkeepsie,  Wheeling,  Memphis,  etc.,  have  been  made  more  than  100  feet  high, 
assembled  and  connected  in  a  horizontal  plane  and  the  first  one  erected  by  simply  lashing  down  the  heel 
and  revolving  it  into  a  vertical  plane  by  two  or  three  carefully  adjusted  lines  on  each  side,  attached  at 
different  points,  and  leading  over  shear  poles  to  the  hoisting  engine.  Of  course  the  lofty  timber  frame  was 
securely  held  by  guys  front  and  back  that  were  kept  constantly  taut.  After  the  first  bent  is  erected  the 
second  one  is  easily  hoisted  from  it,  both  are  braced  together,  and  the  structure  immediately  becomes  stable 
and  rigid. 

In  erecting  an  ordinary  simple  truss  span  the  members  of  the  floor  system  and  lower  chords  are  distrib- 
uted on  the  falsework  in  position  before  the  traveller  lifts  and  assembles  the  other  members,  commencing 
either  at  one  end  and  going  straight  across  or  beginning  at  the  centre  and  working  to  one  end  and  then 
returning  and  v/orking  out  to  the  other  end. 

The  trusses  are  carefully  placed  in  alignment  and  vertical  plane,  but  are  blocked  up  at  the  joints,  where 
they  rest  on  wedges,  to  a  much  greater  camber  curve  than  exists  in  the  completed  structure,  so  as  to  allow 
the  adjustments  for  the  final  connections  being  made  by  driving  out  the  wedges. 

In  suspended  spans  between  cantilevers,  the  end  piers  rest  in  expansion  slots  in  the  cantilever  arms, 
thus  permitting  longitudinal  motion  that  would  allow  the  portions  of  the  centre  trusses  to  hang  down  and 
make  the  final  connection  impossible  unless  special  provision  is  made  for  adjusting  their  length  and  contm!- 
ling  their  position  and  inclination.  This  usually  consists  of,  at  each  of  the  eight  slots,  a  fixed  and  movable 
roller  separated  by  a  wedge  which  is  commanded  by  a  powerful  screw  so  that  by  entering  or  withdrawing 
one  or  both  of  its  wedges  the  extremity  of  any  truss  segment  may  be  raised,  lowered,  protruded,  or 
withdrawn. 

Usually  they  are  set  in  the  beginning  so  as  to  allow  for  the  deflection  of  dead  load  and  traveller,  and 
still  be  inclined  above  the  required  cambered  position,  so  that  the  adjustment  requires  only  the  slacking  off 
of  the  wedges. 


AMERICAN  METHODS  OF  BRIDGE  ERECTION. 


Before  this  device  was  well  known  the  speaker  was  required  to  provide  erection  adjustment  for  the 
second  large  cantilever  bridge  ever  built  in  this  country,  at  St.  Johns,  N.  B.,  and  designed  simple  stirrup 
irons  or  U  bolts  that  engaged  the  movable  top  chord  pins  and  had  screw  ends  passing  through  fixed  bearing 
plates  against  which  their  nuts  rested.  This  afforded  a  tension  adjustment  for  drawing  or  letting  out 
the  top  chord,  while  large  bolts  screwd  through  solid  boxes  in  the  ends  of  the  lower  chords  and  having 
rounded  ends  bearing  in  liemispherical  cups  in  a  casting  bolted  to  the  end  of  the  cantilever  formed  a  com- 
pression adjustment  and  could  be  easily  set  up  or  slacked  off  so  as  to  lengthen  or  shorten  the  lower  chord. 
This  method  proved  simple,  economical,  and  satisfactory. 

In  the  erection  of  the  steel  arch  bridge  spanning  the  Mississippi  River  at  St.  Louis  (see  Plates  XXIV 
and  XXXI)  an  effort  was  made  to  insert  the  closing  twelve-foot  sections  of  the  tubular  ribs  of  the  arches  by 
cooling  them  down  with  ice.  The  ribs  were  covered  with  gunny  sacks  and  ice  was  placed  upon  their  upper 
sides,  but  the  temperature  of  the  tubes  was  lowered  by  this  means  only  28°  F.  in  the  day  and  5°  at  night 
(from  95°  and  62°  respectively).  These  effects  were  not  sufficient  to  allow  tubes  of  the  normal  length  to  be 
inserted.  Other  sections  were  used,  which  were  provided  with  adjustable  sleeve-nuts  with  right  and  left 
threads,  allowing  an  extension  of  in.  These  were  worked  by  means  of  a  powerful  wrench  operating 
upon  a  i^in.  steel  bar  inserted  in  holes  through  the  nut.  The  upper  and  lower  nuts  were  turned 
alternately  as  the  arch  rose  and  fell  with  rising  and  falling  temperatures,  till  the  normal  length  was 
obtained. 

Working  Plant  comprises  travellers,  derricks,  hoisting  engines,  tackle,  pile  drivers,  pumps,  locomo- 
tives, cars,  differential  hoists,  hydraulic  and  screw  jacks,  dynamometers,  steam,  pneumatic,  hydraulic,  and 
electric  appliances  and  hand  tools.  Stationary  derricks  are  much  used  for  unloading  and  handling  material 
and  for  hoisting  secondary  members.  They  are  ordinarily  of  the  familiar  mast  and  boom  pattern,  with 
hollow  vertical  mast  pivot  through  which  the  fall  lines  lead  to  the  hoisting  drums,  and  are  generally  rigged 
with  manilla  running  gear  and  wire-rope  standing  guys.  Sometimes  they  are  stiff  legged,  and  sometimes 
have  also  bases  of  timber  sills,  enabling  them  to  be  easily  moved.  Balanced  derricks  are  sometimes  used. 
At  the  Washington  Bridge  across  the  Harlem  River  all  the  material  for  one  of  the  510  ft.  arches  was 
lifted  to  a  distributing  platform  by  a  balanced  derrick  that  had  twin  overhangs,  a  bearing  on  top  and  a 
friction  collar  and  rollers  at  the  bottom  of  the  trussed  arms.  Small  four-wheeled  trucks  or  "  lauries  ' 
are  usually  used  for  distributing  the  iron  work  on  the  span  and  bringing  it  from  the  yards,  but  these  often 
are  provided  with  special  lifting  devices  for  expediting  loading  and  unloading,  especially  if  there  is  no  yard 
derrick.  A  car  having  two  braced  vertical  posts  in  the  middle,  capped  by  an  overhanging  beam  with  a 
tackle  suspended  from  each  end,  is  very  convenient  for  picking  up  a  pair  of  stringers  and  bringing  them  and 
unloading,  one  on  each  side  of  the  track,  nearly  in  the  required  position.  .\.  very  effective  arrangement  was 
devised  by  a  young  erector,  that  simply  consisted  of  a  long  trussed  beam  with  its  forward  end  somewhat 
elevated  and  carrying  a  tackle  that  was  mounted  close  to  the  front  end  on  a  small  car;  it  could  be  run  out 
in  the  yard  and  the  fall  belayed  and  the  lever  tipped  down  to  hook  on  to  a  heavy  piece ;  then  several  men. 
mounting  the  long  arm  of  the  lever,  would  raise  the  load  and  push  the  car  to  any  required  place,  when  its 
burden  would  be  lowered  by  slacking  off  the  fall  line.  This  device  was  especially  convenient  for  handling 
floor  beams. 

The  many  varieties  of  hoisting  apparatus  in  use  are  generally  some  form  of  multiple  spooled  engine,  with 
capstan  heads  and  drums;  some  of  them  have  eight  spools,  each  driven  by  independent  gearing  and  com- 
manded by  clutches  and  brakes,  so  that  any  one  may  haul  up  or  slack  off  independently,  the  rope  only 
taking  two  or  three  turns  around  the  spool  and  then  being  tailed  off  so  as  to  maintain  the  same  efficiency 
always.  These  engines  usually  have  a  locomotive  gearing  to  enable  them  to  propel  themselves  on  standard- 
gauge  track.  An  ingenious  method  has  been  employed  to  hoist  them  to  the  top  of  high  falsework.  Ac 
the  Poughkeepsie  Bridge  a  six-spool  engine  was  delivered  by  a  boat  underneath  the  traveller,  from  whose 
top  beams,  250  feet  above,  four  sets  of  tackle  were  hung  and  their  lower  blocks  were  hooked  on  to  the 
2ngine-bed,  one  on  each  corner ;  the  fall  line  of  each  was  wrapped  around  a  capstan  head  and  kept  taut  by  a 
man  mounted  on  the  frame.  Another  man  started  the  engine,  and  as  the  fall  lines  were  wound  up  and 
tailed  off,  the  engine  pulled  itself  and  its  five  men  up  to  the  top  of  the  falsework  and  was  let  down  on 
beams  slipped  under  to  receive  it.  Sometimes  a  locomotive  can  be  advantageously  employed  to  hoist  heavy 
members,  as  at  the  Niagara  Railroad  Suspension  Bridge,  where  the  heavy  saddles  were  hoisted  to  the  tops  ol 
the  80- ft.  towers  by  a  line  rove  through  upper  and  lower  snatch-blocks,  and  led  from  the  latter  to  a  loco 
motive  that  simply  steamed  off  with  it  and  drew  the  load  up  after  it.  The  running  tackle  used  should  be 
best  manilla  rope ;  \\,  \\,  and  if  are  the  common  sizes,  generally  rove  through  double  and  treble  blocks  with 
lignum  vitae  or  steel  shelves  16'',  18",  or  20"  in  diameter,  and  steel  hooks  and  cases.  Members  are  generally 
lifted  with  chain  slings  which  must  be  properly  fastened  to  avoid  cross-straining  the  Imks,  which  can  be 
very  easily  snapped.    Special  hook  clamps  are  often  provided  to  fit  the  flanges  of  girders.    For  extra-heavy 


5^6 


APPENDIX. 


strains  a  luff  tackle  may  be  used,  consisting  merely  of  a  second  purchase  attached  to  the  fall  line  of  the  first. 
When  this  is  commanded  by  a  heavily  geared  windlass  or  a  hoisting  engine  great  power  is  developed,  but  is 
slow  and  troublesome  in  its  application. 

Very  heavy  pieces  and  large  masses  or  assembled  structures  may  be  moved  horizontally  on  greased  skids 
by  hydraulic  jacks  that  are  made  of  from  5  to  100  tons  capacity,  and  can  also  raise  or  lower  it,  but  should 
then  be  closely  followed  by  solid  blocking. 

In  coinmencing  to  drive  a  connecting  pin,  the  holes  in  the  different  members  often  do  not  match,  and 
a  square-ended  pin  could  not  be  entered.  Therefore,  the  end  which  is  shouldered  and  threaded  for  a  nut 
receives  a  pilot  of  the  same  diameter  as  the  body  of  the  pin,  up  to  which  it  is  screwed  to  fit  tightly,  while  the 
front  end  is  made  conical  and  can  enter  a  half-hole  into  which  it  is  driven,  drawing  the  pieces  into  position 
and  allowing  the  pin  to  follow  easily.  The  pia  should  always  be  driven  by  a  wooden  maul  or  ram  to  avoid 
battering  its  threads.  An  iron-hooped  beam  or  log  suspended  at  the  pin,  by  its  centre  of  gravity  held  from  a 
considerable  height  and  swung  against  the  pin,  does  excellent  service. 

The  specifications  for  most  large  recent  bridges  have  required  machine  field  riveting  which,  in  this 
country,  has  been  done  by  pneumatic  or  hydraulic  tools  similar  to  those  used  in  the  shops.  At  the  Memphis 
Bridge  the  riveters  for  the  floor  system  were  simply  swung  by  long  ropes ;  at  the  Poughkeepsie  Bridge  they 
were  swung  from  a  trolley  that  ran  on  a  longitudinal  track,  moving  in  a  transverse  arc  and  all  revolving 
about  the  centie  of  a  small  traveller  that  cleared  the  inside  braces  of  the  main  traveller.  At  the  Washington 
Bridge  the  arch  rib  splices  were  riveted  up  by  a  machine  that  hung  from  two  differential  hoists  carried  by  a 
trolley  whose  deeply  grooved  wheels  rolled  on  4-inch  round  bars  that  themselves  rolled  freely  on  the  hori- 
zontal braces.  Electric  field  riveters  are  being  much  used  abroad.  A  recent  pattern  has  a  small  motor  that 
by  reducing  gear  operates  a  screw  piston  which  develops  hydraulic  pressure  for  driving  each  rivet. 

Adjustments  of  tension  members  sometimes  are  required  to  be  made  accurately  and  verified;  this  can  be 
done  by  interposing  an  ordinary  spring  dynamometer  so  as  to  form  a  temporary  link  in  the  connection.  But 
this  often  is  a  needless  refinement,  since  the  proportionate,  if  not  the  actual,  tensions  of  members  can  gener- 
ally be  closely  estimated  after  some  experience  by  striking  them  with  a  hammer.  In  adjusting  over  600  floor- 
beam  suspenders  of  tlie  Niagara  Bridge  the  writer  was  able  to  estimate  the  strain  by  feeling  of  the  ropes 
almost  within  the  limits  of  graduation  of  the  dynamometer. 

In  the  same  bridge  L.  L.  Buck  reinforced  the  original  anchor  chains  by  additional  new  links,  and  accu- 
rately adjusted  the  load  taken  up  by  the  latter  by  the  elongation  produced. 

When  only  a  slight  discrepancy  at  first  exists  between  tension  bars  of  the  same  members  they  will 
usually  adjust  themselves  by  proportionate  elongations;  but  if  the  variation  is  too  great  it  may  be  sometimes 
eliminated  by  heating  both  bars  (as  by  wrapping  them  with  oily  waste  and  igniting  it),  and  allowing  them  to 
set  themselves  in  cooling.  The  attempt  to  shorten  the  steel  ribs  of  the  St.  Louis  arches  by  packing  them 
in  ice  proved  unsuccessful  as  described  above. 

Thus  it  is  seen  that  in  bridge  erection  unforeseen  and  perplexing  contingencies  continually  arise,  and 
must  be  met  by  all  the  resources  of  science  and  mechanics. 

The  equipment  of  a  complete  general  erection  outfit  should  comprise  full  kits  of  carpenters'  and 
blacksmiths'  and  masons'  tools,  portable  forges  (these  may  be  improvised  with  half  barrels  filled  with  clay, 
and  a  large  bellows  pumping  into  another  large  barrel  with  tuyere  pipes  to  three  or  four  of  them  will  furnish 
very  satisfactory  and  economical  blast,  and  may  be  very  convenient  for  such  work  as  riveting  up  buckle-plate 
flooring),  riveters'  outfits,  tool  steel,  plenty  of  steel  spikes,  fitting  bolts,  long  bolts  and  washers,  hand  screws, 
hand  chisels  and  gouges,  ratchets,  reamers,  screw  and  set  wrenches,  large  key  wrenches  with  rings,  pinch 
bars,  crowbars,  hand  hammers,  sledges,  a  50- ft.  and  a  loo-ft.  steel  tape,  rubber  clothing,  lights,  and  the 
more  important  tools,  etc.,  mentioned  above., 

American  bridge  engineers  design  their  structures  with  careful  consideration  for  the  erection  require- 
ments, and  of  the  combination  bridges  lately  built  in  the  far  West,  some,  if  not  many,  have  been  specially  con- 
structed to  afford  facility  for  replacing  the  wooden  compression  members  with  iron  without  disconnecting 
them  or  impairing  the  integrity  of  the  structures. 

A  large  and  increasing  proportion  of  the  bridge  erection  here  to  day  consists  in  the  renewal  of  existing 
structures  where  it  is  almost  invariably  demanded  that  the  traffic  shall  not  be  interrupted.  When  permissible 
the  road  is  usually  moved  to  a  temporary  crossing  on  one  side  of  the  old  structure,  which  is  then  demolished 
and  replaced  unrestrictedly,  but  this  method  is  often  not  practicable  and  numerous  other  expedients  are 
resorted  to;  most  often,  probably,  trestle  work  is  put  up  under  the  span  and  the  track  transferred  to  it,  as  well 
as  the  old  structure  after  it  is  disconnected.  As  soon  as  it  is  removed  the  new  structure  is  assembled  upon 
the  trestles,  everything  being  scrupulously  made  to  clear  the  trains  or  only  used  in  fixed  intervals  between 
them.  Sometimes  the  old  bridge  is  made  to  support  the  new  one  till  the  latter  is  swung  and  self-supporting. 


AMERICAN  METHODS  OF  BRIDGE  ERECTION. 


when  it  in  turn  supports  the  old  one  until  completely  removed.    Sometimes  the  old  structure  is  taken  out 

piecemeal  and  replaced  by  the  new,  or  clamped  to  it  for  certain  periods. 

Some  very  remarkable  achievements  have  been  accomplished  by  L.  L.  Buck,  whose  work  on  the  Railroad 
Suspension  Bridge  at  Niagara  Falls  is  a  masterpiece  of  skill  and  ability.  He,  at  first,  opened  the  anchor 
pits,  disconnected  the  main  cables,  replaced  numerous  corroded  wires,  removed  the  anchors,  replaced  them 
and  added  new  links  and  pins  to  their  chains.  Afterwards  he  replaced  the  whole  suspended  wooden  super- 
structure with  steel  and  iron  floors  and  trusses.  At  another  time  he  removed  portions  of  the  masonry 
towers,  and  put  in  new  stones,  and  finally  he  replaced  the  massive  towers  with  steel  structures  standing  on 
substantially  the  same  foundations,  and  accomplished  all  tiie  difficult  operations  quickly,  cheaply,  and  with- 
out loss  of  life  or  any  serious  interruption  of  traffic. 

The  rapidity  with  which  erection  work  can  be  executed  is  illustrated  by  the  bringing  of  the  material  for 
a  2oo-ft.  railroad  bridge  from  a  storage  yard  looo  feet  away,  and  erecting  it  in  i6  working  hours  after  false- 
work was  ready. 

The  1518-ft.  span  of  the  Cairo  Bridge  was  erected  by  Baird  Brothers  in  six  days;  two  spans  were  erected 
and  the  falsework  and  traveller  twice  put  up  and  moved  in  one  month  and  three  days,  inclusive  of  five  days 
of  idle  time.* 

Bridge  erection  is  subject  to  many  dangers,  and  serious  accidents  appear  to  be  inevitable.  Those  occur- 
ring to  single  individuals  are  often  not  the  fault  of  the  victim,  who  is  frequently  injured  by  an  article 
dropped  by  some  one  else  or  knocked  off  from  the  work  by  some  carelessness.  An  experienced  bridgeman 
seldom  falls  from  a  great  height  through  dizziness  or  missteps,  but  may  do  so  by  carelessly  stepping  on  a  loose 
plank.  Some  terrible  accidents  have  occurred  by  the  collapse  of  falsework,  trestling,  and  travellers,  some  of 
them  perhaps  due  to  derailments  or  breakages  while  hoisting  heavy  pieces,  or  possibly  to  outright  general 
weakness,  but  comparatively  few  are  ever  exactly  determined,  except  where,  as  is  too  often  the  case,  trestles 
are  destroyed  by  floods  scouring  out  the  bottom  underneath  them  or  piling  vast  quantities  of  drift  against 
them,  as  was  the  case  at  Wheeling,  W.  Va.,  and  has  been  frequent  in  the  Ohio  River  bridges. 

No  other  calling  demands  and  receives  the  experience,  courage,  good  judgment,  and  personal  endurance 
displayed  by  the  leaders  and  skilled  workmen  in  bridge  erection.  They  must  construct  the  loftiest  and  most 
difficult  scaffolding  solely  by  their  own  resources,  often  in  remote  and  dangerous  positions,  and  upon  them 
must  handle  and  perfectly  adjust  heavy  girders  and  huge  chords,  etc.,  weighing  perhaps  100,000  lbs.,  while 
subject  to  constant  peril  of  destruction  by  storm  and  flood,  or  they  must  build  great  trusses  in  the  very  path 
of  frequent  express  trains  without  impeding  their  progress  or  prejudicing  their  safety.  Under  such  trying 
circumstances  their  work  is  accomplished  with  a  rapidity  and  accuracy  exceeding  that  in  some  comfortable 
and  well  equipped  shops  and  mechanical  plants,  and  the  great  address  and  faithfulness,  general  integrity  and 
reliability  that  they  exhibit  in  tiieir  difficult  tasks  brings  them  into  deserved  prominence  among  constructive 
workmen.  Tliese  men  are  characteristic  of  our  graiul  nation.  Kee[)ing  pace  with  the  unparalleled  creations 
of  this  generation  of  bridge  designers,  they  have  ai)plied  no  ordinary  engineering  skill  to  the  devising  and 
execution  of  erection  methods  whose  success  is  attested  by  scores  of  inonumental  constructions,  and  the 
absence  of  many  great  disasters. 

*  Perhaps  the  most  remarkable  achievement  on  record  (to  March,  1896)  is  found  in  the  erection  of  three  Pegram- 
truss  spans  (see  p.  67),  of  217  feet  each,  on  the  Union  Pacific  R.  R.  These  three  spans  were  erected,  floor  system  put 
in,  and  ties  and  guard-rails  finished  on  Feb.  8,  i8q6,  the  bridge  gang  having  arrived  on  the  ground  on  the  evening  of 
Jan.  24.  Thus  in  twelve  working  days  the  false-work  was  put  in,  the  traveller  erected,  the  old  Ilowe-truss  spans 
removed,  and  660  feet  of  new  bridge  erected  and  completed.  The  last  span,  not  including  the  floor  system,  was  erected 
in  five  hours  and  twenty  minutes,  the  material  being  brought  from  the  storage  yard,  several  hundred  feet  away.  On 
an  average  70  men  were  employed,  28  of  them  being  on  the  traveller.    Only  one  hoisting-engine  was  used. 


INDEX. 


PAGB 

Abutment  reactions  determined  analytically  15-17 

graphically   22 

for  swing  bridges  139,  182 

Abutments,  aesthetic  design  of  419 

Adjustment  of  members    516 

iEsthetic  design,  general  principles  412-414 

influence  of  color  on  415 

material  on  414 

shades  and  shadows  on .  416 

of  the  roadway   421 

substructure  418-420 

superstructure   421 

ornamentation   416-418 

plates  and  comments  422-426 

;iEsthetics,  development  of   411 

Anchor-bolts  267,  309 

Anchorage  of  columns  160,  403,  405,  463 

stand-pipes  429-431 

suspension  bridges   177 

specification  for   487 

Angle-blocks  for  Howe  trusses   351 

Arch  and  equilibrium  polygon  203 

of  three  hinges,  computation  of  stresses   206 

equilibrium  polygon  for    205 

graphic  analysis  of   41 

maximum  stress  in  any  member  205 

of  two  hinges,  computation  of  stresses   211 

deflection  of    208 

equilibrium  polygon  for   2og 

maximum  stress  in  any  member  210 

stresses  due  to  distortion   212 

temp,  changes. . .  211 

with  fixed  ends,  computation  of  stresses   216 

equilibrium  polygon  for   213 

maximum  stress  in  any  member  215 

stresses  due  to  distortion   218 

temp,  changes..  217 

Arches,  advantages  of   241 

analysis  of  203-218 

deflection  formulae  derived  206-208 


erection  of  

kinds  of   203 

Artistic  analysis  412-414 

design.    See  ^Esthetic  design. 

Baltimore  truss,  analysis  of,  for  uniform  loads   60 

Batten  plates,  design  of  253.  341 

Beam,  bending  moment  in  a  31^  45-47 

moment  of  resistance  at  rupture   125 


PACE 

Beam,  shear  and  bending  moment,  relation  between  .  131 


shearing  stress  in  a  47,  134-163 

stress  distribution  in  a   124 

trussed,  analysis  of  a   161 

Beams,  arrangement  in  tall  buildings   440 

curved,  deflection  of  206-208 

deflection  formulae  for   131 

economical  spacing  of   441 

moment  formulae  for   132 

moments  and  shears,  table  of   329 

theory  of  119-142 

elementary  principles  in  120-124 

history  of   119 

Bearing  plates.    See  Pin  plates. 

Bed-plates  for  plate  girders   309 

trestle-columns   400 

specification  for   496 

Bollman  truss,  history  of   7 

Bolts,  use  of   267 

Bottom  chords,  design  of  end   280 

specification  for   490 

Bowstring  truss,  analysis  for  uniform  loads  64-67 

Braced  arches.    See  Arch. 

Bridge,  economical  location  for  a   233 

number  of  spans   235 

Bridge  erection,  accidents  in   517 

defined   508 

details  of   516 

equipment  necessary   516 

falsework  for  511-514 

hoisting  apparatus  for   515 

long  span  bridges   511 

pontoon  method  of   512 

rapidity  of   517 

simple  trusses   514 

viaducts   509 

without  stopping  traffic   516 

working  plant  for  514-516 

Bridge  floors,  design  of  281-284 

discussion  of  237-239 

highway  344-347 

plate  girder  237-239,  311 

Bridge  strains,  apparatus  for  measuring   230 

Bridge  truss,  design  of  a.     See  Railway  bridge. 

economical  dimensions  of  a   321 

Bridge  trusses,  analysis  for  uniform  loads  43-72 

wheel  loads  73-100 

apex  loads   45 

Bridges,  advantages  of  different  forms  236-241 


520 


INDEX. 


Bridges,  aesthetic  design  of.    See  ^Esthetic  design, 
highway.    Sec  Highway  bridges. 
Howe  truss.    See  Howe  truss. 

live  loads  on   45 

long  span,  form  of .  . .    241 

proper  structure  for  given  crossing   234 

railway.    See  Railway  bridges. 

specifications  for  485-500 

styles  and  determining  conditions  233-241 

swing.    See  Swing  bridges. 

weights  of,  formulae  for  . . .  .43,  233,  238-241,  344 
wind  pressure  on   45 

Building  construction.  See  Tall  building  construction, 
also  Mill  building  construction. 

Burr  truss,  history  of   5 

Cable,  deflection  of  172-175 

stress  in,  for  uniform  load   167 

when  stays  are  used   176 

Calking   432 

Camber,  amount  of  291 

of  swing  bridges   379 

purpose  of     226 

specification  for   490 

Cantilever  bridges,  advantages  and  disadvantages,  198,241 

analysis  of  198-202 

concentrated  loads  on   199 

erection  of  511,  514 

forms  of   197 

Forth  bridge   197 

Indiana  and  Kentucky  bridge  ....  200 

influence  lines  for   199 

Kentucky  river  bridge  197 

wind  stresses  in   202 

Centrifugal  force,  effect  of   117 

specification  for   487 

Chord  packing  in  Howe  trusses   352 

Chord  splices    "      "         "    349 

Chord  stresses  for  uniform  loads  49,  62 

wheel  loads  '  79,  85,  88 

Chords,  definition  of   3 

stiff  bottom  280,  490 

top.    See  Top  chord. 

Clearance  for  rivet  heads   258 

specification  for   486 

Clevises,  dimensions  of  standard   249 

Collision  struts  for  Howe  trusses   352 

specification  for   ...  486,  490 

use  of   281 

Color,  artistic  use  of   415 

Columns,  anchorage  of  160,  403,  405,  463-465 

arrangement  of,  in  tall  buildings  440 

bases  for  400,  467 

batter  of  393,  487 

caps  for   .400,  458 

design  of,  for  tall  buildings  450-453 

effect  of  end  conditions   147 

Euler's  formula  for   146 

failure  by  bending   145 

crushing   143 

crushing  and  bending   144 

for  elevated  railroads  ijg,  403-407 


Columns  for  elevated  tanks  434-437 

mill  buildings  463-467,  469-474 

tall       "   440,442-444.450-453 

trestles  393,  397,  400 

forms  of  251,  452 

formulae  for   I43-I53 

loads  on,  in  buildings   442 

new  formula  for   ...  148 

Rankine's  formula  for   141 

straight  line    "       "   151 

stresses  in  159,  405,  450,  463,  469-474 

tests  of   152 

timber   353 

wind  stresses  in  159,  463,  469-474 

See  also  Compression  members. 

Combined  stresses,  action  of   154 

compression  and  bending  . . .  .157,  159 

examples  of   154 

formulae  for   154 

in  end  posts,  from  wind   159 

tension  and  bending   155 

Compound  sections  for  tension  members   251 

Compression  and  bending  157-159 

Compression  members,  bending  due  to  weight   490 

economy  in  manufacture  ... .  251 
resisting  stress . .  252 

end  details  of   253 

forms  of  251,  452 

formulae  for   255 

latticing  of   255 

limiting  dimensions  of . .  .253,  490 

specifications  for  490,  494 

tie-plates  on  253,  341 

See  also  Columns  and  Top  chord. 

Concentrated  loads,  analysis  for  73-100 

conventional  methods   of  treat- 
ment 101-108 

graphical  method  of  analysis.  .  .84-88 

on  cantilever  bridges   igg 

position  of,  for  maximum  floor- 
beam  load   76 

position  of,  for  maximum  mo- 
ment  74 

position  of,  for  maximum  shear  .  77 

Continuous  girders,  for  mulse  for  moments  137— t-jo 

shears   14c 

reaction  constants  for  two  equal 

spans  139,  182 

See  also  Swing  bridges. 

Conventional  methods  of  analysis   101-108,  142 

Coping  of  pier,  aesthetic  design  of   419 

Corbels  in  timber  trestles   383 

Cornices   422 

Counterbrace,  defined   3 

Counters,  defined   4 

Couple,        "    II 

Cranes,  columns  and  girders  for   466 

Crescent  truss  38-41 

Crushing  strength  of  columns,  defined   143 

Dead  loads,  bridges   43 

Ipuildings   443 


INDEX. 


521 


PAGE 

Dead  loads,  highway  bridges   343 

railway  bridges   321 

roofs   33 

specification  for   487 

swing  bridges     377 

Deck-bridge,  defined   5 

Deflection,  effect  of  clianging  height  of  truss   224 

errors  from  neglecting  web   227 

formula  for  a  Pratt  truss  222-224 

for  beams  I3I-I33 

from  chord  and  web   219,  223 

inelastic  =   225 

numerical  computation  of   227 

of  framed  structures  219-227 

suspension  bridges  •■ .  .172-175 

a  Warren  girder   219 

Derrick,  A   510 

Derricks,  description  of   S^S 

Details  of  construction  257-291 

Dimensions,  artistic  consideration  of   413 

Double  intersection  truss,  inclined  chords   70 

parallel      "   57,  97 

triangular  truss   59 

Dowels  for  Howe  trusses  352 

Draw  bridges.    See  Swing  bridges. 

Eccentric  loading  of  members  164,  453 

trusses  115-117 

Eccentricity  of  pins,  effect  of   158 

top  chord,  computation  of   325 

Elastic  limit,  defined   2 

Elasticity,  modulus  of    2 

Elevated  railroads,  characteristic  features  of  402 

columns,  anchorage  for  403,  405 

stresses  in  159,  405-407 

cost  of   410 

economical  span  for   410 

erection  of   510 

expansion  joints  for   278,  404 

illustrations  of  407-409 

lateral  and  longitudinal  stability 

of   403 

live  loads  for   402 

wind  pressure  on   404 

Elevated  tanks,  concentration  of  loads  to  columns  . . .  434 

cost  compared  to  stand-pipes   437 

dimensions  of   437 

general  design  434 

material  for   438 

riveting  of   434 

roof   436 

thickness  of  plates   434 

tower,  design  of   437 

Embankment,  cost  compared  to  iron  trestle   396 

End-lifts  for  swing  bridges  364-370.  380 

Engine  excess   loi 

Engine  for  swing  bridges    374,  381 

Equations  of  equilibrium,  analytical  application  of. .  14-20 
graphical         "         "  .  .21-32 

Equilibrium,  defined   11 

laws  of,  applied  to  framed  structures  .  .11-32 
Equilibrium  polygon,  for  an  arch  of  three  hinges   205 


PAGB 

Equilibrium  polygon,  for  an  arch  of  two  hinges   209 

forces  nearly  parallel   22 

parallel     25 

reactions  found  by  •. .  22 

relation  to  stresses  in  arches.. .  203 

rhrough  three  points   24 

Equivalent  uniform  load  method  of  analysis.  .102,  105,  106 

Excess  loads  loi,  102 

Expansion  joints  for  elevated  railroads   404 

girders  278 

External  forces  defined   i 

Eyebars,  dimensions,  table  of   246 

loss  of  strength  in  246,  248 

manufacture  of   246 

packing  of,  on  pins   273 

parallelism  of   337 

specification  for   495 

stiffened   280 

stress  due  to  weight  of  155,  157 

Facing,  method  of   504 

Falsework,  framed  trestle   512 

pile  "    513 

suspension  bridge  for   511 

trussed  span  for   512 

Fatigue  formula   243 

of  metals   242 

Fink  truss,  history  of   7 

Flange  areas,  formula  for   298 

Flange  plates,  length  of   301 

Flange  splices   307 

Flanges,  design  of   301 

spacing  of  rivets  in   298 

transference  of  stress  to  304-306 

Floor,  weight  of    443 

Floor-beam  concentration,  computation  of  76,  83 

Floor-beam  hangers  at  hip,  design  of  336 

plate     284 

specification  for   493 

stiffened   280 

stresses  in   168 

Floor-beams  for  a  highway  bridge,  design  of   346 

Howe  truss   353 

railway  bridge   327 

Floor  system  for  a  highway    "   345,347 

railway      "   326-330 

through  plate  girders  237-239,  311 

timber  trestles   383 

general  design  of  237-239,  281-284 

Floor  systems,  illustrations  of  282-283 

Forces,  kinds  of   11 

Form  of  a  structure  413 

Forth  bridge,  diagram  of   197 

wind  pressure  experiments  at   33 

Foundations  in  soft  soil  444-447 

Framed  structures,  classification  of   508 

defined   i 

erection  of  508-517 

Friction  coefficient,  specification  for  490 

Friction  in  swing  bridges  372,  375 

Gin-pole,  defined   509 


522 


INDEX. 


PAGE 

Gin-pole,  raising  of   510 

use  of   509 

Girders,  moments  and  shears,  table  of   ...  329 

pin  bearing  for   278 

See  plate  girders,  also  lattice  girders. 

Graphic  analysis  for  concentrated  loads  83-88 

of  trusses  21-42,  68-71 

Gravity  lines  non-intersecting,  stresses  due  to  163 

Hand  rails   344 

Hangers  at  hip  336 

plate  284 

specification  for  493 

stiffened   280 

stresses  in   168 

tests  of     484 

Highway  bridge,  design  of  floor  system  for  a  345-347 

lateral  bracing  for  a   348 

main  members     "    347 

Highway  bridges,  dead  loads  on  343 

design  of  details   344 

dimensions  of  344 

floors,  forms  of   344 

forms  of   345 

general  conditions   343 

hand  rails  for   344 

live  loads  on   44,  343 

panel  length  for   345 

weights  of  44,  343 

working  stresses  for  344 

Hip  vertical,  design  of   336 

stiffened   280 

Historical  development  of  the  truss  5-10 

theory  of  beams   120 

Hoisting  engines   515 

Howe  truss,  analysis  for  uniform  loads  54-56 

angle  blocks  and  bearing  plate   351 

bill  of  material  for  a   355 

chord  splices  in  349-351 

design  of  a  345-356 

history  of  the.   7 

miscellaneous  details    352 

Howe  trusses,  advantages  and  use  of  349 

of  long  span,  details   356 

weights  and  quantities  43,  354 

working  stresses  for   353 

I-beam  spans,  specifications  for. ...   491 

Indiana  and  Kentucky  bridge,  analysis  of  ,  200 

Influence  lines,  defined   73 

for  cantilever  bridges   199 

double  intersection  trusses   96 

moment  74,  88 

panel  load   76 

shear   77 

skew  bridges   100 

inspection  of  mill-work   502 

shop- work  504-507 

records  for  504-506 

specification  for  498 

Iron  and  steel,  aesthetic  qualities  of  414 

inspection  of   502 


rACB 

Iron  and  steel,  manufacture  o(   501 

wrought,  specification  for.  ,  497 

Joints,  details  of,  for  railway  bridges  330-341 

packing  of  289,  291 

riveted.    See  Riveted  joints. 

Kentucky  River  bridge   .  197 

Lateral  bracing,  analysis  of  109-118 

effect  on  hip  verticals   336 

for  plate  girders   309 

railway  bridges  328,  342 

roof  trusses  319,  465 

swing  bridges  376 

forms  of  282-288 

specifications  for   496 

Lateral  rod,  stress  due  to  weight  of   157 

Lattice  bars,  design  of,  for  a  railway  bridge   342 

general  design  of   255 

size  of  291,  494 

Lattice  girders,  advantages  of   239 

design  of   239 

erection  of   509 

pin  bearing  for    278 

secondary  stresses  in  163 

transportation  of   508 

weight  of  43,  240 

Lattice  truss,  analysis  of   59 

Launhardt's  formula   243 

Lenticular  truss,  analysis  of   67 

Live  loads  for  bridges  44,  79,  101-108,  142,  343 

buildings   444 

elevated  railroads   302 

roofs   33 

Load  line   85 

Lomas  nuts   268 

Long  span  bridges,  forms  of   241 

Loop  eyes,  dimensions  of   247 

manufacture  and  use  of   250 

Masonry,  pressure  on,  for  swmg  bridges   359 

specification  for  496 

Material,  influence  of,  on  aesthetic  appearance   414 

kind  of,  for  various  members  486 

specifications  for   497 

Maxwell  diagrams  27-29 

Memphis  bridge,  movable  bearing  for   277 

Method  of  sections.   19 

Mill  buildings,  analysis  for  wind  stresses  469-474 

analysis,  methods  of   461 

bracing,  systems  of   465 

columns,  bases  for   467 

horizontal  forces  on  463-465 

connections,  design  of  467 

corrugated  sheeting  for  468 

girders  466 

lighting  and  ventilation  468 

loads  on,  horizontal  461 

vertical  460 

piers  for    467 

purlins  for  468 


mDEX. 


5*5 


PAGE 

Mill  buildings,  roof  trusses  466,  468 

track  rails    467 

types  of   460 

wind  pressure  on   461 

Mill-work  described   501 

inspection  of   502 

Minimum  sections,  specifications  for  490 

Modulus  of  elasticity,  definition  and  examples   2 

Moment,  defined    11 

in  an  arch      204 

in  a  beam,  for  uniform  loads  31,  45 

concentrated  loads  45,  75 

cantilever  bridge   199 

truss,  for  uniform  loads    32 

concentrated  loads  75,  88 

influence  lines  for  74,  88,  96,199 

sign  of   45 

Moment  of  inertia,  graphical  method  of  finding   128 

of  top-chord  section  325 

tabular  computation  of   ....  127 

Moments  and  shears  on  girders,  table  of   329 

Moments  of  inertia,  formulae  for   122 

Moments  of  resistance  at  rupture   125 

formulae  for  122 

Moments,  tabulation  of  wheel  load   79 

Nuts,  dimensions  of   268 

Ornamentation,  distribution  of   416 

form  of   417 

object  of  413,  416 

of  coping  of  pier   419 

of  roadway   422 

quantity  of.   416 

scale  of   417 

style  of   418 

Packing  of  joints  289,  291 

for  a  railway  bridge  332-338 

Painting,  specification  for  499 

Panel  concentrations,  computation  of   87 

in  a  Whipple  truss   97 

Panel  length  for  highway  bridges    345 

Panel  loads   45,  87 

Panels,  long  versus  short   321 

Pedestals  for  iron  trestles     39g 

Pegram  truss,  analysis  for  uniform  loads  67-69 

wheel       "   91-94 

Petit  truss,  advantages  of   240 

analysis  for  uniform  loads   69 

wheel        "   9-1.  95 

Piers  and  abutments,  aesthetic  design  of  418-420 

Pile  trestles    See  Trestles. 

Pin  bearing  for  girders   278 

Pin-connected  trusses,  advantages  of   240 

secondary  stresses  in   165 

weight  of   241 

Pin-joints,  deflection  caused  by  play  in   225 

Pin-plates,  design  of,  for  a  railway  bridge  338-340 

for  tension  members   251 

length  of   338 

thickness  of  330,  332-338 


PACE 

Pin,  position  of,  in  top  chord   325 

Pins,  bearing  values  of   273 

bearing  values  of  plates  on,  diagram  of   274 

bending  moments  on,  calculation  of  270-271 

table  of   272 

design  of,  for  a  railway  bridge  '332-338 

dimensions  of  standard   26C 

driving  of   51C 

eccentricity  of   158 

fibre  stresses  in   271 

for  lateral  rods  ,  269 

grip,  calculation  of   269 

packing  of  eyebars  on    272 

shearing  stresses  in   269 

specifications  for  489,  496 

stresses,  calculation  of. ...   269-278 

working  stresses  for  271-273,  489 

Pivot  for  deck  swing  bridges   359 

through  swing  bridges   357 

Plate  girder  swing  bridges   357 

end  lifts  for   365 

Plate  girders,  advantages  of   236 

anchor  bolts  for   309 

bed-plates  for   309 

calculation  by  moment  diagram   84 

concentrated    loads,    transference  to 

web  297-303 

design  of  deck  299-310 

through  310-31 1 

economical  depth  of   299 

erection  of  509,  512 

expansion  bearings  for   309 

expansion  joints  for   278 

flange  angles,  size  of     302 

flange  plates,  length  of   301 

flanges,  design  of   298,  301 

resultant  stress  on  rivets  306 

riveting  of  299,  304-307 

splices  in   307 

transference  of  web  stress.  304-306 

floors,  design  of    311 

forms  of   238 

forms  of   237 

lateral  bracing  of   309 

long,  specifications  for   494 

moment  for  concentrated  loads   75 

resisted  by  web   292 

moments  and  shears  on,  table  of  329 

pin  bearing  for   278 

shear  for  concentrated  loads   77 

specifications  for   492-494 

stiffeners,  design  of  136,  297,  308 

theoretical  treatment  of  292-298 

transportation  of   508 

web,  design  of   302 

dimensions  of   293 

splice  for  moment  and  shear  .293-297 

splices,  rivets  in   307 

weight  of  43,  238 

width  of   309 

Plates,  bearing  value  of,  on  rivets   274 

limiting  weight  of   293 


PAGE 

Pontoons  used  in  erection   512 

Pony  truss,  defined   5 

Portal  bracing,  analysis  of  110-114,  159,  288,  405-407 

design  of  288,  455 

for  Howe  trusses   352 

in  tall  buildings  455-457 

specification  for   496 

Portal  strut,  design  of  329 

Post  splices  in  timber  trestles  391 

Post  truss,  analysis  of   59 

history  of   9 

Posts,  details  of   340 

Pratt  truss,  advantages  of   240 

analysis  for  uniform  loads   56 

wheel       "   79-^3 

deflection  formula  for  222-224 

from  web  and  chord  223,  224 

depths  for  maximum  stiffness   224 

history  of   9 

subdivided.     See   Baltimore  truss,  also 
Petit  truss. 

Punched  holes,  effect  of,  on  steel  and  iron  476-483 

Punching,  methods  of   503 

Purlins,  design  of.  316,  468 

spacing  of   314 

Queen-post  truss  without  counters   161 

Railings,  aesthetic  design  of  422 

Railway  bridge,  design  of  a  321-342 

floor  system  326^330 

joint  details  330~34i 

lateral  bracing  328,  342 

main  members  323-326 

pins  332-338 

shoes    341 

Stress  sheet  for   331 

tabulation  of  stresses  in  a   323 

working  stresses  for  a   322 

Ram  and  toggle-joint  end  lift  368-370 

Reaming,  methods  of   503 

tests  to  determine  effect   480 

Redundant  members,  stresses  in  220,  222,  226-230 

Repeated  stress   242 

Resultant,  defined   11 

of  concurrent  forces  11-13 

non-concurrent  forces   13 

Reversed  stress,  experiments  on   242 

Rivet  heads,  clearance  for   258 

size  and  weight  of   257 

Rivet  spacing,  general  rules  ...    258 

in  angles,  channels,  and  beams   266 

girder  flanges  298,  304-307 

specifications  for   491 

Riveted  joints,  calking  of   432 

design  of  262,  263 

kinds  of   261 

strength,  table  of   432 

tests  of    263-265,  477-481 

Riveted  truss.    See  Lattice  girder. 

Riveting,  clearance  for  tools  259 

effect  of,  on  strength  of  net  section  482 

field   516 


Mes 

Riveting,  inspection  of   506 

methods  of   503 

of  elevated  tanks   434 

stand-pipes  431 

Rivets,  bending  of    261 

conventional  signs  for   267 

in  pin-plates  338-340 

length  or  grip  of   259 

location  of,  on  drawings   266 

material  for   257 

secondary  stresses  in. ... ,   165 

size  of  257-258 

specifications  for  491 

strength,  tables  of   260 

Roadway,  aesthetic  design  of  421 

Rods,  dimensions  of  square  and  round   247 

specifications  for  495 

use  of  c   248 

Roller-bearing,  design  for  heavy..,   277 

Rollers,  bearing  strength  of  273,  275-277 

experiments  on  275-277 

for  swing  bridges,  conical  359 

design  of   379 

Roof  coverings,  kinds  of  314 

weight  of  33,  317 

Roof-trusses,  analysis  of  34-42,  314,  469-474 

arch  truss   41 

crescent  truss  38-41 

dead  load  on   33 

design  of   317-320 

economical  spacing  of   315 

erection  of   509 

expansion  of  468 

Fink,  analysis  of   36 

table  of  coefficients  for   318 

forms  of  35,  313,  466 

lateral  bracing  of  319,  465 

live  load  on  33,  314,  460 

purlins,  design  of  316,  468 

proportions  of   313 

reactions  for   34 

riveted  versus  pin-connected   313 

weight  of     314 

wind  pressure  on  33,  314,  461-463,  465 

working  stresses  for  315 

Screw-threads  on  steel  rods   483 

Secondary  stresses,  defined   163 

due  to  eccentric  loading   164 

non  -  intersecting  gravity 

lines   163 

rigidity  of  joints   165 

in  rivets   165 

Shades  and  shadows,  artistic  value  of   416 

Shear  in  an  arch   204 

a  beam  47,  77 

a  truss    77 

cantilever  bridges   200 

sign  of   47 

Shoe,  design  of  a   289 

Shoes,  details  of   341 

specifications  for  496 


INDEX. 


PAGE 

Shop- work,  inspection  of    504-507 

processes  described  502-504 

Single  shapes  for  tension  members   250 

Skew  bridges,  analysis  for  uniform  loads   71 

wheel       "    99 

Skew  portals   114 

Sleeve-nuts,  dimensions  of   249 

Spandrel  sections,  design  of  447-450 

Specifications  for  bridges  485-500 

steel  and  iron  438,  497 

working  stresses   322 

Stand-pipes,  anchorage  for  429-431 

bottom  of   432 

calking  of  432 

capacity  of   427 

cost  of   437 

material  for  428,  438 

ornamentation  of    433 

riveting  of  431 

thickness  of  plate  428 

top  of  432 

wind  pressure  on  438 

Stays,  action  of  169-174 

deflection  of  I73-I75 

stresses  in    176 

Steel  and  iron,  inspection  of  501,  502 

manufacture  of    501 

punching,  effect  of  484 

cost  of,  compared  to  iron   476 

for  stand-pipes  and  tanks  428,  438 

medium,  workmanship  on  496 

punching  of  476-483 

reaming,  tests  to  determine  effect   480 

sheared  edges  on   481 

soft,  use  of,  in  bridges  475-485 

specifications  for  438,  497 

tests  showing  effect  ot  punching  477-481 

bars,  screw  threads  on   4S3 

versus  wrought-iron  475-485 

Stiffeners  for  stringers  and  plate  girders.  136,  297,  308,  326 

Stiffening  truss,  action  of   168 

deflection  of....  168,  174 

stresses  in  169-171 

with  stays   176 

Stone,  aesthetic  qualities  of   414 

Straightening    501 

Strain,  defined   I 

Strain  diagrams   123 

Strains,  apparatus  for  measuring   230 

Stress  and  strain   I 

defined    i 

Stress  sheet,  defined   330 

for  a  railway  bridge     331 

specification  for   486 

Stresses,  calculation  of,  specification  for  487 

sign  of   ...  3 

tabulation  of  computation   55 

Stringers  and  floor-beams,  specifications  for  493 

for  Howe  trusses   353 

railway  bridges ...    326 

timber  trestles   383 

lateral  bracing  for  284 


5*5 

PAGB 

Stringers  minimum  depth   281 

moment  and  shear  for  wheel  loads  83,  329 

spacing  of,  specification  for   485 

stiffeners  for  ,   .  326 

weight  of   326 

Struts,  defined   3 

Style   412 

Sub-panels,  methods  of  making   279 

Substructure,  aesthetic  design  of  418-430 

Sub-struts  for  supporting  compression  members  281 

Suspension  bridges,  anchorage   177 

as  falsework   511 

cable  167,  172 

erection  of    511 

hangers  173-176 

history  of   166 

stays   173-176 

stiffening  truss  168-171,  173-176 

theory  of     167-178 

Sway  bracing,  analysis  of.  114-116,  159,405,455,  463,  469 

design  of   288 

for  elevated  railroads  405 

mill-buildings  463,  469 

tall        "    455 

timber  trestles   384 

Swing  bridge,  design  of  a  376-381 

Swing  bridges,  analysis  of  179-196 

bearing  on  masonry   359 

camber  of   379 

centre  bearing,  analysis  of  183-192 

conical  rollers  for  centre  bearing   359 

dead  load    '83,377 

deflection  of   379 

design  of  357-38i 

end  lifts  364-370,  380 

engine,  design  of  374,  381 

erection  of   511 

floor  system,  design  of   376 

forms  of  333,  357,  360 

formula;  for  reactions  179-181 

friction  constants   375 

friction  in  turning   372 

inertia  of   373 

lateral  bracing  for   376 

lift   195 

lifting  of,  power  required   371 

machinery  for  operating  370 

methods  of  supporting  at  the  centre. .  362 

pivot  for  centre  bearing   357 

plate  girder   357 

pony  truss   358 

power  required  in  lifting. . .  .367-371,  373 

turning   373 

proper  form  to  use  ■   333 

reaction  constants  for  139,  182,  186 

reactions,  formulje  for   179-181 

resistances  to  turning  371-373 

rim  bearing,  four  supports,  analysis  of  193 
rim  bearing,  four  supports,  equal  loads 

on  the  turntable,  analysis  of   195 

rim  bearing,  four  supports,  equal  mo- 
ments at  centre,  analysis  of   193 


INDEX. 


PAGE 

Swing  bridges,  rim  bearing,  three  supports,  analysis 


of   194 

rollers,  conical,  for  centre  bearing...  359 

turning,  arrangements  for  370,  380 

resistances  to  37i~373 

turntable,  design  of  377-379 

weight  of  183,  377 

wind  pressure  in  turning   372 

wind  stresses  in   ig6 

with  variable  moments  of  inertia   196 

work  done  in  lifting  and  turning.  -371-373 

Symmetry,  importance  of   412 

Tall  buildings,  beams,  arrangement  of  440 

calculation  of   450 

spacing  of   441 

columns,  arrangement  of  440 

calculation  of  450-453 

forms  of   452 

loads  on  442-444 

design  of   440-459 

details  457-459 

floor,  type  of   443 

foundations  444-447 

iron  and  steel  in  439 

loads  in   442 

procedure  in  designing  440 

spandrel  sections  447-450 

wind  bracing  453-457 

Tanks.    See  Elevated  tanks. 

Temperature  stresses  in  arches  211,  217 

Tension  and  bending  154,  155 

Tension  members,  adjustment  of   516 

compound  sections  for   251 

design  of  245-251 

eyebars   245 

rods   248 

single  shapes   250 

specifications  for  495 

Tests  on  columns   152 

riveted  joints  263,  477 

rollers   275 

Thames  River  bridge,  friction  of   375 

Three  moments,  equation  of   137 

Through  bridge  defined   5 

Through  plate  girder,  design  of   310 

Thrust  in  an  arch   204 

Tie  plates,  design  of    253,  341 

Ties  defined    3 

Ties,  railroad,  specification  for  485 

Timber  trestle.    See  Trestles. 

Timber,  working  stresses  for   353 

Toggle-joint  end-lift  368-370 

Top  chord  as  stringer   158 

centre  of  gravity  of   325 

design  of   324 

eccentric  loading,  effect  of   158 

joints,  design  of   289 

position  of  pin  in   325 

splices,  design  of   340 

stress  due  to  weight  of  157-159 

Towers  for  elevated  tanks  437 

Travellers,  construction  of  510,  514 


rAGE 

Trestles,  iron,  bracing  of   397 

columns,  bases  for   400 

batter  of   393 

caps  for   400 

connections  for  400 

design  of   397 

cost,  compared  to  embankment   396 

economic  length  of  span    394 

erection  of   510 

general  design  of  391 

lateral  stability  of   393 

length  of  tower  span   394 

stresses  in  397-399 

weight  of   395 

wind  pressure  on   393 

specifications  for  485-500 

timber,  bents   382 

conditions  of  use   382 

corbels   383 

erection  of   511 

floor  systems  for  ,  383 

for  falsework  512-514 

illustrations  of  standard  385-390 

joints  in   382 

longitudinal  stability  of   384 

Stringers  for   383 

splices  in  391 

sway  bracing  for  384 

working  stresses  for   391 

Triangular  truss  51,  61 

Triple-intersection  truss   59 

Truss,  action  of  a   4 

defined   3 

historical  development  of  the  5-10 

for  suspension  bridges.    See  Stiffening  truss. 

Truss  members,  design  of  242-256 

Trussed  beam,  analysis  of.   161 

Trusses,  deflection  of  219-227 

economy  of  various  forms   278 

forms  forswmg  bridges  360-362 

See  also  Bridge  trusses  and  Roof  trusses. 

Turn-buckle  instead  of  sleeve-nut   250 

Turning  arrangements  for  swing  bridges   380 

Turntable,  design  of  a  377-379 

Turntable,  method  of  loading  362-364 

Upset  screw  ends,  dimensions  of   248 

Viaducts,  erection  of   510 

See  Elevated  railroads,  also  Trestles. 

Warren  girder,  advantages  of   240 

analysis  for  uniform  loads  51-54 

wheel       "    88 

Water  towers,  kinds  of  427 

See  Stand-pipes,  also  Elevated  tanks. 

Watertown  Arsenal  tests  of  columns   152 

riveted  joints  263,  477 

Web,  distribution  of  load  over  297,  303 

thickness  of   302 

Web  members  defined   3 

Web  splices  293-297,  307 

Web  stiffeners  136,  297,  308 


INDEX. 


527 


PAGE 

Web  stresses  for  wheel  loads  81-83,  86,  90 

inclined  chords  62-64 

parallel      "   50,  51 

Weight  ol  floors  443 

highway  bridges  44,  363 

Howe  trusses  43,  354 

lattice  girders   43,  240 

pin-connected  trusses  43,  241 

plate  girders  43,  238 

rivet  heads   257 

roof  coverings   317 

roof  trusses ...    314 

stringers   326 

swing  bridges   183 

trestles,  iron   395 

Weyrauch's  formula   244 

Wheel  loads,  tabulation  of  moments   79 

See  a/w  Concentrated  toads. 

Whipple  truss,  analysis  for  uniform  loads   57 

wheel       "   97-99 

history  of   8 


Wind  bracing.    See  Lateral  bracing.  Portal  bracing, 
also  Sway  bracing. 


PAGE 

Wind  loads,  specification  for  487 

Wind  pressure,  experiments  on   33 

on  bridges   45 

buildings   461-463 

elevated  railroads   404 

roofs  ,   33 

stand-pipes   429 

trestles   393 

Wind  stresses  in  bridges  109-114,  159,  196,  202 

buildings  455,469-474 

elevated  railroads  405-407 

roof  trusses   315 

Work,  equation  of  external  and  internal  219 

Working  formulae  243-245 

Working  plant  for  erection  514-516 

Working  stresses  for  highway  bridges   344 

railway       "    322 

roof  trusses   315 

timber   353 

specifications  for   488 

Wrought-iron  compared  to  soft  steel  475-485 

specifications  for  497 


